Digitized by the Internet Archive in 2011 with funding from Research Library, The Getty Research Institute http://www.archive.org/details/firstsixbooksofeOObyrn C'€>r>-'^ ~? 'ih BYRNE'S EUCLID THE FIRST SIX BOOKS OF liTHE ELEMENTS OF EUCLID WITH COLOURED DIAGRAMS AND SYMBOLS .V' » THE FIRST SIX BOOKS OF THE ELEMENTS OF EUCLID IN WHICH COLOURED DIAGRAMS AND SYMBOLS ARE USED INSTEAD OF LETTERS FOR THE GREATER EASE OF LEARNERS BY OLIVER BYRNE SURVEYOR OF HER MAJESTY'S SETTLEMENTS IN THE FALKLAND ISLANDS AND AUTHOR OF NUMEROUS MATHEMATICAL WORKS LONDON WILLIAM PICKERING 1847 TO THE RIGHT HONOURABLE THE EARL FITZWILLL\M, ETC. ETC. ETC. THIS WORK IS DEDICATED BY HIS LORDSHIPS OBEDIENT AND MUCH OBLIGED SERVANT, OLIVER BYRNE. INTRODUCTION. HE arts and fciences have become fo extenfive, that to faciUtate their acquirement is of as much importance as to extend their boundaries. Illuftration, if it does not fhorten the time of ftudy, will at leaft make it more agreeable. This Work has a greater aim than mere illuftration ; we do not intro- duce colours for the purpofe of entertainment, or to amufe by certain combinations of tint and form, but to airift the mind in its refearches after truth, to increafe the facilities of inflrudlion, and to diffufe permanent knowledge. If we wanted authorities to prove the importance and ufefulnefs of geometry, we might quote every philofopher fmce the days of Plato. Among the Greeks, in ancient, as in the fchool of Peftalozzi and others in recent times, geometry was adopted as the befl: gymnaftic of the mind. In facfl, Euclid's Elements have become, by common confent, the bafis of mathematical fcience all over the civilized globe. But this will not appear extraordinary, if we confider that this fublime fcience is not only better calculated than any other to call forth the fpirit of inquiry, to elevate the mind, and to ftrengthen the reafoning faculties, but alfo it forms the beft introdudlion to moft of the ufeful and important vocations of human life. Arithmetic, land-furveying, men- furation, engineering, navigation, mechanics, hydroftatics, pneumatics, optics, phyfical aftronomy, &c. are all depen- dent on the propolitions of geometry. viii INTRODUCTION. Much however depends on the firft communication of any fcience to a learner, though the beft and moft eafy methods are feldom adopted. Propofitions are placed be- fore a ftudent, who though having a fufficient underftand- ing, is told juft as much about them on entering at the very threfliold of the fcience, as gives him a prepolleffion moft unfavourable to his future ftudy of this delightful fubjedl ; or " the formalities and paraphernalia of rigour are fo oftentatioufly put forward, as almoft to hide the reality. Endlefs and perplexing repetitions, which do not confer greater exactitude on the reafoning, render the demonftra- tions involved and obfcure, and conceal from the view of the ftudent the confecution of evidence." Thus an aver- fion is created in the mind of the pupil, and a fubjeft fo calculated to improve the reafoning powers, and give the habit of clofe thinking, is degraded by a dry and rigid courfe of inftrudlion into an uninterefting exercife of the memory. To raife the curiofity, and to awaken the liftlefs and dormant powers of younger minds fliould be the aim of every teacher ; but where examples of excellence are wanting, the attempts to attain it are but few, while emi- nence excites attention and produces imitation. The objedl of this Work is to introduce a method of teaching geome- try, which has been much approved of by many fcientific men in this country, as well as in France and America. The plan here adopted forcibly appeals to the eye, the moft fenlitive and the moft comprehenfive of our external organs, and its pre-eminence to imprint it fubjedl on the mind is fupported by the incontrovertible maxim exprefled in the well known words of Horace : — Segnius irritant animos demijfa per auran ^uam qua fimt oculis fuhjeSla fidelibus. A feebler imprefs through the ear is made, Than what is by the faithful eye conveyed. INTRODUCTION. ix All language confifts of reprefentative figns, and thole figns are the befl which efFedl their purpofes with the greateft precifion and difpatch. Such for all common pur- pofes are the audible figns called words, which are ftill confidered as audible, whether addreffed immediately to the ear, or through the medium of letters to the eye. Geo- metrical diagrams are not figns, but the materials of geo- metrical fcience, the objedt of which is to Ihow the relative quantities of their parts by a procefs of reafoning called Demonftration. This reafoning has been generally carried on by words, letters, and black or uncoloured diagrams ; but as the ufe of coloured fymbols, figns, and diagrams in the linear arts and fciences, renders the procefs of reafon- ing more precife, and the attainment more expeditious, they have been in this inflance accordingly adopted. Such is the expedition of this enticing mode of commu- nicating knowledge, that the Elements of Euclid can be acquired in lefs than one third the time ufually employed, and the retention by the memory is much more permanent; thefe facts have been afcertained by numerous experiments made by the inventor, and feveral others who have adopted his plans. The particulars of which are few and obvious ; the letters annexed to points, lines, or other parts of a dia- gram are in fadt but arbitrary names, and reprefent them in the demonftration ; inftead of thefe, the parts being differ- ently coloured, are made g to name themfelves, for their forms incorrefpond- ing colours represent them in the demonftration. In order to give a bet- ter idea of this fyftem, and A of the advantages gained by its adoption, let us take a right X INTRODUCTION. angled triangle, and exprefs fome of its properties both by colours and the method generally employed. Some of the properties of the right angled triangle ABC, expreffed by the method generally employed. 1 . The angle BAC, together with the angles BCA and ABC are equal to two right angles, or twice the angle ABC. 2. The angle CAB added to the angle ACB will be equal to the angle ABC. 3. The angle ABC is greater than either of the angles BAC or BCA. 4. The angle BCA or the angle CAB is lefs than the angle ABC. 5. If from the angle ABC, there be taken the angle BAC, the remainder will be equal to the angle ACB. 6. The fquare of AC is equal to the fum of the fquares of AB and BC. The fame properties expreffed by colouring the different parts. That is, the red angle added to the yellow angle added to the blue angle, equal twice the yellow angle, equal two right angles. -^ + A = Or in words, the red angle added to the blue angle, equal the yellow angle. ▲ <^H^ CZ JK^ or The yellow angle is greater than either the red or blue angle. INTRODUCTION. xl iL 4. jl^^ or Either the red or blue angle is lefs than the yellow angle. ^^^^^ minus ^HL In other terms, the yellow angle made lefs by the blue angle equal the red angle. That is, the fquare of the yellow line is equal to the fum of the fquares of the blue and red lines. In oral demonftrations we gain with colours this impor- tant advantage, the eye and the ear can be addreffed at the fame moment, fo that for teaching geometry, and other linear arts and fciences, in clafTes, the fyftem is the beft ever propofed, this is apparent from the examples juft given. Whence it is evident that a reference from the text to the diagram is more rapid and fure, by giving the forms and colours of the parts, or by naming the parts and their colours, than naming the parts and letters on the diagram. Befides the fuperior limplicity, this fyftem is likewife con- fpicuous for concentration, and wholly excludes the injuri- ous though prevalent pradlice of allowing the ftudent to commit the demonftration to memory ; until reafon, and fadl, and proof only make impreffions on the underftanding. Again, when ledluring on the principles or properties of figures, if we mention the colour of the part or parts re- ferred to, as in faying, the red angle, the blue line, or lines, &c. the part or parts thus named will be immediately feen by all in the clafs at the fame inftant ; not fo if we fay the angle ABC, the triangle PFQ^the figure EGKt, and fo on ; xii INTRODUCTION. for the letters mufl be traced one by one before the fludents arrange in their minds the particular magnitude referred to, which often occafions confufion and error, as well as lofs of time. Alfo if the parts which are given as equal, have the fame colours in any diagram, the mind will not wander from the objedl before it ; that is, fuch an arrangement pre- fents an ocular demonftration of the parts to be proved equal, and the learner retains the data throughout the whole of the reafoning. But whatever may be the advantages of the prefent plan, if it be not fubftituted for, it can always be made a powerful auxiliary to the other methods, for the purpofe of introdudlion, or of a more fpeedy reminifcence, or of more permanent retention by the memory. The experience of all who have formed fyftems to im- prefs fadts on the underftanding, agree in proving that coloured reprefentations, as pidlures, cuts, diagrams, &c. are more eafily hxed in the mind than mere fentences un- marked by any peculiarity. Curious as it may appear, poets feem to be aware of this fadl more than mathema- ticians ; many modern poets allude to this viiible fyftem of communicating knowledge, one of them has thus expreffed himfelf : Sounds which addrefs the ear are loft and die In one fhort hour, but thefe which ftrilce the eye, Live long upon the mind, the faithful fight Engraves the knowledge with a beam of light. This perhaps may be reckoned the only improvement which plain geometry has received fince the days of Euclid, and if there were any geometers of note before that time, Euclid's fuccefs has quite eclipfed their memory, and even occalioned all good things of that kind to be alfigned to him ; like ^Efop among the writers of Fables. It may alfo be worthy of remark, as tangible diagrams afford the only medium through which geometry and other linear INTRODUCTION. xiii arts and fciences can be taught to the blind, this vifible fys- tem is no lefs adapted to the exigencies of the deaf and dumb. Care muft be taken to fliow that colour has nothing to do with the lines, angles, or magnitudes, except merely to name them. A mathematical line, which is length with- out breadth, cannot poffefs colour, yet the jundtion of two colours on the fame plane gives a good idea of what is meant by a mathematical line ; recolledt we are fpeaking familiarly, fuch a jundlion is to be underftood and not the colour, when we fay the black line, the red line or lines, &c. Colours and coloured diagrams may at firfl: appear a clumiy method to convey proper notions of the properties and parts of mathematical figures and magnitudes, how- ever they will be found to afford a means more refined and extenfive than any that has been hitherto propofed. We fliall here define a point, a line, and a furface, and demonflrate a propofition in order to fhow the truth of this affertion. A point is that which has pofition, but not magnitude ; or a point is pofition only, abftradled from the confideration of length, breadth, and thicknefs. Perhaps the follow- ing defcription is better calculated to explain the nature of a mathematical point to thofe who have not acquired the idea, than the above fpecious definition. Let three colours meet and cover a portion of the paper, where they meet is not blue, nor is it yellow, nor is it red, as it occupies no portion of the plane, for if it did, it would belong to the blue, the red, or the yellow part; yet it exifts, and has pofition without magnitude, fo that with a Uttle refledlion, this June- XIV INTRODUCTION. tion of three colours on a plane, gives a good idea of a mathematical point. A line is length without breadth. With the afliftance of colours, nearly in the fame manner as before, an idea of a line may be thus given : — Let two colours meet and cover a portion of the paper; where they meet is not red, nor is it blue ; therefore the jundlion occu- pies no portion of the plane, and therefore it cannot have breadth, but only length : from which we can readily form an idea of what is meant by a mathematical line. For the purpofe of illuftration, one colour differing from the colour of the paper, or plane upon which it is drawn, would have been fufficient ; hence in future, if we fay the red line, the blue line, or lines, &c. it is the junc- tions with the plane upon which they are drawn are to be underftood. Surface is that which has length and breadth without thicknefs. When we confider a folid body (PQ), we perceive at once that it has three dimenfions, namely : — length, breadth, and thicknefs ; fuppofe one part of this folid (PS) to be red, and the other part (QR) yellow, and that the colours be diflinft without commingling, the blue furface (RS) which feparates thefe parts, or which is the fame S thing, that which divides the folid without lofs of material, mufl be without thicknefs, and only poffeffcs length and breadth ; INTRODUCTION. XV this plainly appears from reafoning, limilar to that juft em- ployed in defining, or rather delcribing a point and a line. The propofition which we have felefted to elucidate the manner in which the principles are applied, is the fifth of the firft Book. In an ifofceles triangle ABC, the internal angles at the bafe ABC, ACB are equal, and when the fides AB, AC are produced, the exter- nal angles at the bafe BCE, CBD are allb equal. Produce _i__ and make ■■■■ "^ Draw ^— — and (B. i.pr. 3.) and and common and ^ = -^ (B. I. pr. 4.) Again in >^ and N. t ^ xvi INTRODUCTION. and ^ = ^; and ^^^ ^ ^^^ (B. i. pr. 4). But C^E. D. By annexing Letters to the Diagratn. Let the equal fides AB and AC be produced through the extremities BC, of the third Tide, and in the produced part BD of either, let any point D be afllimed, and from the other let AE be cut off equal to AD (B. i. pr. 3). Let the points E and D, fo taken in the produced fides, be con- nedted by ftraight lines DC and BE with the alternate ex- tremities of the third fide of the triangle. In the triangles DAC and EAB the fides DA and AC are refpedlively equal to EA and AB, and the included angle A is common to both triangles. Hence (B i . pr. 4.) the line DC is equal to BE, the angle ADC to the angle AEB, and the angle ACD to the angle ABE ; if from the equal lines AD and AE the equal fides AB and AC be taken, the remainders BD and CE will be equal. Hence in the triangles BDC and CEB, the fides BD and DC are refpedively equal to CE and EB, and the angles D and E included by thofe fides are alfo equal. Hence (B. i. pr. 4.) INTRODUCriON. xvii the angles DBC and ECB, which are thofe included by the third fide BC and the productions of the equal fides AB and AC are equal. Alfo the angles DCB and EBC are equal if thofe equals be taken from the angles DCA and EBA before proved equal, the remainders, which are the angles ABC and ACB oppofite to the equal fides, will be equal. Therefore in aii ifofceles triangle y &c. Q^E. D. Our object in this place being to introduce the fyftem rather than to teach any particular fet of propofitions, we have therefore feledled the foregoing out of the regular courfe. For fchools and other public places of infi:rud:ion, dyed chalks will anfwer to defcribe diagrams, &c. for private ufe coloured pencils will be found very convenient. We are happy to find that the Elements of Mathematics now forms a confiderable part of every found female edu- cation, therefore we call the attention of thofe interefiied or engaged in the education of ladies to this very attractive mode of communicating knowledge, and to the fucceeding work for its future developement. We fhall for the prefent conclude by obferving, as the fenfes of fight and hearing can be fo forcibly and infiianta- neously addreffed alike with one thoufand as with one, the million might be taught geometry and other branches of mathematics with great eafe, this would advance the pur- pofe of education more than any thing that might be named, for it would teach the people how to think, and not what to think ; it is in this particular the great error of education originates. XVlll THE ELEMENTS OF EUCLID. BOOK I. DEFINITIONS. I. A point is that which has no parts. II. A line is length without breadth. III. The extremities of a line are points. IV. A ftraight or right line is that which lies evenly between its extremities. V. A furface is that which has length and breadth only. VI. The extremities of a furface are lines. VII. A plane furface is that which lies evenly between its ex- tremities. VIII. A plane angle is the inclination of two lines to one ano- ther, in a plane, which meet together, but are not in the fame diredlion. IX. ^ A plane redlilinear angle is the inclina- ^r tion of two ftraight lines to one another, ^^^ which meet together, but are not in the ir fame flraight line. BOOK I. DEFINITIONS. XIX When one ftraight line Handing on ano- ther ftraight Hne makes the adjacent angles equal, each of thefe angles is called a rigkf angle, and each of thefe lines is faid to be perpendicular to the other. A XI. An obtufe angle is an angle greater than a right angle. XII. An acute angle is an angle lefs than a right angle. XIII. A term or boundary is the extremity of any thing. XIV. A figure is a furface enclofed on all fides by a line or lines. XV. A circle is a plane figure, bounded by one continued line, called its cir- cumference or periphery ; and hav- ing a certain point within it, from which all ftraight lines drawn to its circumference are equal. XVI. This point (from which the equal lines are drawn) is called the centre of the circle. XX BOOK I. DEFINITIONS. XVII. A diameter of a circle is a ftraight line drawn through the centre, terminated both ways in the circumference. XVIII. A femicircle is the figure contained by the diameter, and the part of the circle cut off by the diameter. XIX. A fegment of a circle is a figure contained by a ftraight line, and the part of the cir- cumference which it cuts off. ^•••••••* ••'•' XX. A figure contained by ftraight lines only, is called a redli- linear figure. XXI. A triangle is a redlilinear figure included by three fides. XXII. A quadrilateral figure is one which is bounded by four fides. The fi:raight lines ■^— «— . and .^_«— i«> connecfting the vertices of the oppofite angles of a quadrilateral figure, are called its diagonals. XXIII. A polygon is a redilinear figure bounded by more than four fides. BOOK I. DEFINITIONS. XXI XXIV. A triangle whofe three fides are equal, is faid to be equilateral. XXV. A triangle which has only two fides equal is called an ilbfceles triangle. XXVI. " A fcalene triangle is one which has no two fides equal. XXVII. A right angled triangle is that which has a right angle. XXVIII. An obtufe angled triangle is that which has an obtufe angle. XXIX. An acute angled triangle is that which has three acute angles. XXX. Of four-fided figures, a fquare is that which has all its fides equal, and all its angles right angles. XXXI. A rhombus is that which has all its fides equal, but its angles are not right angles. XXXII. u An oblong is that which has all its angles right angles, but has not all its fides equal. xxii BOOK L POS'lVLATES. XXXIII. A rhomboid is that which has its op- pofite fides equal to one another, but all its fides are not equal, nor its angles right angles. XXXIV. All other quadrilateral figures are called trapeziums. XXXV, ^^—--^,^g„^^^ Parallel ftraight lines are fuch as are in ^^^^^^^^^^ the fame plane, and which being pro- duced continually in both directions, would never meet. POSTULATES. I. Let it be granted that a flraight line may be drawn from any one point to any other point. II. Let it be granted that a finite ftraight line may be pro- duced to any length in a ftraight line. III. Let it be granted that a circle may be defcribed with any centre at any diflance from that centre. AXIOMS. I. Magnitudes which are equal to the fame are equal to each other. II. If equals be added to equals the fums will be equal. BOOK I. AXIOMS. xxiii III. If equals be taken away from equals the remainders will be equal. IV. If equals be added to unequals the fums will be un- equal. V. If equals be taken away from unequals the remainders will be unequal. VI. The doubles of the fame or equal magnitudes are equal. VII. The halves of the fame or equal magnitudes are equal. VIII. Magnitudes which coincide with one another, or exactly fill the fame fpace, are equal. IX. The whole is greater than its part, X. Two ftraight lines cannot include a fpace. XI. All right angles are equal. XII. If two ftraight lines ( } meet a third ftraight line ( ) fo as to make the two interior angles ( and jj^ ) on the fame fide lefs than two right angles, thefe two ftraight lines will meet if they be produced on that fide on which the angles are lefs than two right angles. XXIV BOOK I. ELUCIDATIONS. The twelfth axiom may be expreffed in any of the fol- lowing ways : 1 . Two diverging ftraight lines cannot be both parallel to the fame flraight line. 2. If a ftraight line interfeft one of the two parallel ftraight lines it mufl alfo interfedt the other. 3. Only one ftraight line can be drawn through a given point, parallel to a given ftraight line. Geometry has for its principal objefts the expofition and explanation of the properties oi figure, and figure is defined to be the relation which fubfifts between the boundaries of fpace. Space or magnitude is of three kinds, linear, fuper- ficial, ■Si.w^foUd. Angles might properly be confidered as a fourth fpecies of magnitude. Angular magnitude evidently confifts of parts, and muft therefore be admitted to be a fpecies ol quantity The ftudent muft not fuppofe that the magni- tude of an angle is affefted by the length of the ftraight lines which include it, and of whofe mutual divergence it is the mea- fure. The vertex of an angle is the point where \}[\& fides or the legs of the angle meet, as A. An angle is often defignated by a fingle letter when its legs are the only lines which meet to- gether at its vertex. Thus the red and blue lines form the yellow angle, which in other fyftems would be called the angle A. But when more than two B lines meet in the fame point, it was ne- ceflary by former methods, in order to avoid confufion, to employ three letters to defignate an angle about that point. BOOK I. ELUCIDATIONS. xxv the letter which marked the vertex of the angle being always placed in the middle. Thus the black and red lines meeting together at C, form the blue angle, and has been ufually denominated the angle FCD or DCF The lines FC and CD are the legs of the angle; the point C is its vertex. In like manner the black angle would be defignated the angle DCB or BCD. The red and blue angles added together, or the angle HCF added to FCD, make the angle HCD ; and fo of other angles. When the legs of an angle are produced or prolonged beyond its vertex, the angles made by them on both fides of the vertex are faid to be vertically oppofite to each other : Thus the red and yellow angles are faid to be vertically oppofite angles. Superpojition is the procefs by which one magnitude may be conceived to be placed upon another, fo as exadlly to cover it, or fo that every part of each fhall exadly coin- cide. A line is faid to be produced, when it is extended, pro- longed, or has its length increafed, and the increafe of length which it receives is called its produced part, or its produSlion. The entire length of the line or lines which enclofe a figure, is called its perimeter. The firft fix books of Euclid treat of plain figures only. A line drawn from the centre of a circle to its circumference, is called a radius. The lines which include a figure are called \isjides. That fide of a right angled triangle, which is oppofite to the right angle, is called the hypotenufe. An oblong is defined in the fecond book, and called a reSlangle. All the lines which are confidered in the firfl: fix books of the Elements are fuppofed to be in the fame plane. The Jiraight-edge and compajfcs are the only inflruments. xxvi BOOK I. ELUCIDATIONS. the ufe of which is permitted in Euclid, or plain Geometry. To declare this reflridlion is the objedl of the pojiulates. The Axioms of geometry are certain general proportions, the truth of which is taken to be felf-evident and incapable of being eftabliflied by demonftration. Propojitions are thofe refults which are obtained in geo- metry by a procefs of reafoning. There are two fpecies of propofitions in geometry, problems and theorems. A Problem is a propofition in which fomething is pro- pofed to be done ; as a line to be drawn under fome given conditions, a circle to be defcribed, fome figure to be con- rtrudled, &c. Th.t folution of the problem confifts in fhowing how the thing required may be done by the aid of the rule or ftraight- edge and compafTes. The demonftration confifts in proving that the procefs in- dicated in the folution really attains the required end. A Theorem is a propofition in which the truth of fome principle is afi^erted. This principle mufl: be deduced from the axioms and definitions, or other truths previously and independently ellabliihed. To fhow this is the objedl of demonftration. A Problem is analogous to a poftulate. A Theorem refembles an axiom. A Pojlulate is a problem, the folution of which is afiiimed. An Axiom is a theorem, the truth of which is granted without demonftration. A Corollary is an inference deduced immediately from a propofition. A Scholium is a note or obfervation on a propofition not containing an inference of fufiicient importance to entitle it to the name of a corollary. A Lemma is a propofition merely introduced for the pur- pofe of eftabliftiing fome more important propofition. xxvu SYMBOLS AND ABBREVIATIONS. ,*, exprefles the word therefore. *,' becaufe. zz equal. This fign of equaHty may be read equal to, or is equal to, or are equal to ; but any difcrepancy in regard to the introdudlion of the auxiliary verbs Is, are, &c. cannot affedl the geometri- cal rigour. ^ means the fame as if the words ' not equal' were written. r~ fignifies greater than. ^ . . . . lefs than. Cjl . . . . not greater than. j] . . . . not lefs than. -\- is vtzdplus [fjiore), the fign of addition ; when interpofed between two or more magnitudes, fignifies their fum. — is read minus {lefs), fignifies fubtracftion ; and when placed between two quantities denotes that the latter is to be taken from the former. X this fign exprefi"es the produdl of two or more numbers when placed between them in arithmetic and algebra ; but in geometry it is generally ufed to exprefs a rect- angle, when placed between " two flraight lines which contain one of its right angles." A reBangle may alfo be reprefented by placing a point between two of its conterminous fides. : :; : exprefies an analogy or proportion ; thus, if A, B, C and D, reprefent four magnitudes, and A has to B the fame ratio that C has to D, the propofition is thus briefly written, A : B ; : C : D, A : B = C : D, A C °'"b = d. This equality or famenefs of ratio is read, xxviii STMBOLS AND ABBREVIAnONS. as A is to B, fo is C to D ; or A is to B, as C is to D. II fignifies parallel to. J_ . . . . perpendicular to. . angle. . right angle. CIS two right angles, ^1^ or I N briefly defignates a point. C =, or ^ fignifies greater, equal, or lefs than. The fquare defcribed on a line is concifely written thus. In the fame manner twice the fquare of, is expreffed by 2 \ def. fignifies definition. pos pofiulate. ax axiom. hyp hypothefis. It may be necefiary here to re- mark, that the hypothefis is the condition aflumed or taken for granted. Thus, the hypothefis of the pro- pofition given in the Introduction, is that the triangle is ifofceles, or that its legs are equal. conft confiruElion. The confiruBion is the change made in the original figure, by drawing lines, making angles, defcribing circles, &c. in order to adapt it to the argument of the demonfi:ration or the folution of the problem. The conditions under which thefe changes are made, are as indisputable as thofe con- tained in the hypothefis. For infi:ance, if we make an angle equal to a given angle, thefe two angles are equal by conftrudlion. Q^E. D ^lod erat detnonfirandum. Which was to be demonftrated. CORRIGENDA. xxix Faults to be correEied before reading this Volu7Jie. Page 13, line 9, /or def. 7 read ^z.L 10. 45, laft line, /or pr. 19 r^^^ pr. 29. 54, line 4 from the bottom, /or black and red line read blue and red line. 59, line 4, /or add black line fquared read add blue line fquared. 60, line 17, /or red line multiplied by red and yellow line read red line multiplied by red, blue, and yellow line. 76, line 11, for def. 7 read dt?. 10. 81, line lOyfor take black line r^i2ii take blue line. 105, line 11, for yellow black angle add blue angle equal red angle read yellow black angle add blue angle add red angle. 129, laft line, /or circle read triangle. 141, line I, /or Draw black line read Draw blue line. 196, line 3, before the yellow magnitude infert M. (Euclib. BOOK I. PROPOSITION I. PROBLEM. N a given finite firaight line ( ) to dejcribe an equila- teral triangle. Defcribe I "^^ and o (postulate 3.); draw and — (poft. i.). then will \ be equilateral. (def. 15.); — (def. 15.), • ^_ -mm — (axiom, i .) ; and therefore \^ is the equilateral triangle required. Q^E. D B BOOK I. PROP. II. PROB. ROM aghenp'jhit ( ■■ ), to draic ajiraight line equ.al to a green finite firaight line ( ). Draw — — — — (poil. I.), defcribe Afpr. I.), produce — — (poll. o 2.), defcribe (poft. 3.), and (poll. 3.) ; produce — ^— "" (port. 2.), ther is the line required. For and (def. 15.), (conll.), .*. (ax. 3.), but (def. 15.' drawn from the given point ( is equal the given line Q. E. D. BOOK I. PROP. in. PROP. ROM the greater ( "—) of tivo given Jiraight lines, to cut off a part equal to the kfs ( ). Draw (poll:. 3 .), then (pr. 2.) ; defcribe For and (def. 15.), (conll.) ; (ax. I.). Q. E. D. BOOK I. PROP. IF. THEOR. F two triangles have two fides of the one refpeSlively equal to two fdes of the other, ( I to ' and ^__ to w^^m. ) and the angles { and ^ ) contained by thofe equal fdes alfo equal ; then their bafes or their fdes (-^-^— and ^^^^) are alfo equal : and the remaining and their remain- ing angles oppofte to equal fides are refpeSlively equal ( ^^ =: ^^ and ^^ n ^^ ) ; and the triangles are equal in every refpeB. Let the two triangles be conceived, to be fo placed, that the vertex of the one of the equal angles. or fliall fall upon that of the other, and with then will ^^— to coincide coincide with » i if ap- plied: confequently will coincide with or two flraight lines will enclofe a fpace, which is impoffible (ax. lo), therefore and ^=» ^ ^^ , and as the triangles * = > A-^ coincide, when applied, they are equal in every refpedl. Q. E. D. BOOK I. PROP. V. THEOR. N anj ifofceles triangle A if the equal Jides be produced, the external angles at the bafe are equal, and the internal angles at the bafe are alfo equal. Produce and y (poft. 2.), take — - = 9 (pr- 3-); draw -i^— — » and n . Then in both, and A A / \ and / \ we have, = (conft.), A common to (hyp.) /. Jk = and ^ = ^ (pr. 4.). ^ = ^ and 1^^ -zz ^^ \ and ^^» ^ ^^ (pr. 4.) but ^ = ^ "*' Jk = JL ^'-^' Q. E. D. BOOK I. PROP. Ft. THEOR. A and N any triangle ( / \ ) ;/' two angles ( ' and ^L ) are equal, the Jides ( ■— ■ ■~ ) oppofite to them are alfo equal. For if the fides be not equal, let one of them I — ■ be greater than the other and from it cut off (pr. 3.), draw- Then (conft.) m A.naA, (hyp.) anc common, ,*. the triangles are equal (pr. 4.) a part equal to the whole, which is abfurd ; ,*, neither of the fides — "» or ' is greater than the other, /. hence they are equal Q^E. D. BOOK I. PROP. FII. THEOR. N the fame bafe (■ ■), a7id on the fa}7ie Jide of it there cannot be tivo triangles having their conterminous fides ( and — ^— ^ •— — ■ and «i^i— ii^—) at both extremities of the bafe, equal to each other. When two triangles ftand on the fame bale, and on the fame iide of it, the vertex of the one Ihall either fall outlide of the other triangle, or within it ; or, laftly, on one of its lides. llructed fo that #='' If it be poffible let the two triangles be con- '«■ rzzzz — zizzz f ^^^" draw ——---- and, = ^ (Pr- 5-) .'. ^^ ^ ^^ and but (pr. 5.) yf = ^^ therefore the two triangles cannot have their conterminous which is abfurd. fides equal at both extremities of the bafe. Q. E. D. BOOK I. PROP. Fill. THEOR. F two triangles have two Jides of the one refpec- tjvely equal to two Jides of the other and .—m^ =r ), and alfo their bafes ( ^ •— ), equal ; then the and ) angles ( contained by their equal Jides are alfo equal. If the equal bafes and be conceived to be placed one upon the other, fo that the triangles fhall lie at the fame fide of them, and that the equal fides «______ and .i.....i_ , —«-.—. and _____ be con- terminous, the vertex of the one mufi: fall on the vertex of the other ; for to fuppofe them not coincident would contradidl the laft propofition. Therefore the fides cident with and . , and ., being coin- k-k Q. E. D. BOOK I. PROP. IX. PROP. Take O bifeB a given reSlilinear angle {^ J. (PJ*- 3-) draw , upon which defcribe ^^ (pr. i.). draw Becaufe _ = ..^... (confl.) and ^^^— common to the two triangles and (conft.). 4 = (pr. 8.) Q. E. D. 10 BOOK I. PROP. X. PROB. O i>tye^ a given finite Jlraight line [f^^^mmmmwm'^. and common to the two triangles. Therefore the given line is bifefted. Q;E. D. BOOK L PROP. XL PROB. II ( : a perpendicular. ROM a given point ( I ), in a given Jlraight line — ), to draw Take any point (• cut off ) in the given line, (pr- 3-)' / \ (Pr. I.), conftrudl draw — — and it fliall be perpendicular to the given line. For (conft.) (conft.) and - common to the two triangles. Therefore ^|| z:z. J. (pr. 8.) (def. 10.). C^E.D. 12 BOOK I. PROP, XII. PROB. O draw a Jlraight line perpendicular to a given / indefinite Jlraight line (^^^ ^ from a given [point /ys. ) "without. With the given point /|\ as centre, at one fide of the line, and any diftance — ^^— capable of extending to the other fide, defcribe Make draw ^ (pr. 10.) and then For (pr. 8.) lince (conft.) and common to both, = (def. 15.) and (def. 10.). Q. E. D. BOOK I. PROP. XIII. THEOR. 13 HEN a Jlralght line ( ..m^^m^ ) Jlanding upon another Jlraight line ( ) makes angles with it; they are either two right angles or together equal to two right angles. If be _L to gf..A=C£^ then, (def. 7.). But if draw be not _L to — — — J. ;(pr. II.) (conft.). Q. E. D. H BOOK I. PROP. XIV. THEOR. F two Jiraight lines fneeting a thirdjlraight line (i ' ), at the fame pointy and at oppofite Jides of it, make with it adjacent angles and A ) egual to two right angles ; thefe fraight lines lie in one continuous Jiraight line. For, if pofTible let and not be the continuation of then + but by the hypothefis 4 = ^ + (ax. 3.) ; which is abfurd (ax. 9.). , is not the continuation of and the like may be demonftrated of any other flraight line except , ,*, ^-^— is the continuation of Q. E. D. BOOK I. PROP. XV. THEOR. 15 gles and F two right lines ( and ■' ' I ) interfe£t one another, the vertical an- and ^ are equal. ► - <* ► 4 In the fame manner it may be fliown that Q^E. D. i6 BOOK I. PROP. XVI. THEOR. F a fide of a is produced, the external trian- greater than either of the internal remote angles ( ▲ .A )• Make Draw — (pr. lo.). - and produce it until ■^^— ; draw — ^— • , In and #•••' ► 4 and (conft. pr. 15.), /. ^m = ^L (pr. 4.), ...f^.A. In like manner it can be fhown, that if •—-■•• be produced, ^^^^ Q ^^k , and therefore is [= ^ii. Q. E. D. which is ^z BOOK I. PROP. XVII. THEOR. 17 NY tivo angles of a tri- angle f * are to- gether lefs than two right angles. Produce + then will ^Oi But, mik [= Mk (pr- 16.) and in the fame manner it may be Ihown that any other two angles of the triangle taken together are lefs than two right angles. Q;E. D. i8 BOOK I. PROP. XVIIL THEOR. A N any triangle if one Jide vbm* be greater than another •^^mmmm-^ ^ the aUgk Of- pojite to the greater Jide is greater than the angle oppoftte to the lefs. 1. e. ^ Make Then will (pr. 3.), draw A.A (pr- 5-) J but i£k (pr. 16.) and much more IS ^-> Q. E. D. BOOK I. PROP. XIX. THEOR. 19 A F m any triangle one angle J/j^ be greater than another ^^^ the Jide which is oppojite to the greater angle, is greater than the Jide oppojite the lefs. If be not greater than or then muft If then which is contrary to the hypothefis. — is not lefs than •^■— ^—j for if it were, which is contrary to the hypothefis : Q. E. D. 20 BOOK I. PROP. XX. THEOR. NY two fides and iBMMH of a triangle Z\ taken together are greater than the third fide ( ). Produce and make >< (pr- 3-); draw Then becaufe ------ ^ (conft.). (ax. 9.) + and ,*, + (pr. 19.) Q.E.D BOOK I. PROP. XXL THEOR. 21 F from any point ( / ) within a triangle ' Jlraight lines be drawn to the extremities of one fide ( ), thefe lines mujl he toge- ther lefs than the other two fdes, but muJl contain a greater angle. Produce + add to each. (pr. 20.), + + (ax. 4.) In the fame manner it may be fhown that ... + [Z + which was to be proved. 4 ■.A (pr. 16.), (pr. 16.), Q^E.D. 22 BOOK I. PROP. XXII. THEOR. \IVE'N three ng/it lines < -■••— the fum of any two greater than the third, to conJlru6i a tri- angle whofe Jides Jhall be re- fpeSlively equal to the given lines. ■■■•■«a««^«M AfTume Draw — — ^ and -^— • s: With defcribe and and I (pr. 2.). as radii, (poft. 3.); draw and then will For and ■ be the triangle required. "' i Q. E. D. BOOK I. PROP. XXIII. PROB. 23 iT a given point ( ) in a given Jiraight line (^^^»»— ■), to make an angle equal to a given re 51 i lineal angle (.^^j^ )• Draw — — — . between any two points in the legs of the given angle. Conftruct v (pr. 22.) fo that — ^^^ = . and Then jgj^ = ^J^ (pr. 8.). Q. E. D. 24 BOOK I. PROP. XXir. THEOR. X> F two triangles have two fides of the one refpec- tively equal to twofdes of the other ( to and ------ to ), and if one of A the angles ( <3. .\ ) contain- ed by the equal fdes be greater than the other (c.»«^), the fide ( ^-^-^^ ) isohich is oppofte to the greater angle is greater than thefde ( - . . . ) which is oppofte to the lefs angle. Make and — L^ - ly (pr. 23.), = (pr- 3-). draw ..-••-■-•» and -■——■. Becaufe ^— — ^ 3: — •— — (ax. i. hyp. conft.) but and .*. ^ = ^ (F- but (pr. 19.) (pr.4.) Q. E. D. BOOK I. PROP. XXV. THEOR. 25 F two triangles have two fides ( " '■ " ■' and ) of the one refpeBively equal to two fides ( and — — ) of the other, but their bafes unequal, the angle fubtended by the greater bafe (««—■—■) of the one, muji be greater than the angle fubtended by the lefs bafe ("■"■■"*•) of the other. ^Im- ^ , C or H] ^^ ^^^ is not equal to ^^ ^^ •=. ^^ then ^^^^ := — — i- (pr. 4.) for if which is contrary to the hypothefis ; ^H^ is not lefs than ^^ for if A :ti A then i "H ' (pr. 24.), which is alfo contrary to the hypothefis : /.A [= A. Q^E. D. 26 BOOK I. PROP. XXVL THEOR. Case I. F two triangles have two angles of the one re- fpedlively equal to two angles of the other. ( and Case II. tf) Let y), and a fide of the one equal to afde of the other fmilarly placed with refpeSl to the equal angles, the remaining fdes and angles are refpeSlively equal to one another. CASE I. and I which lie between the equal angles be equal, then -^— — ^ ^^— ■••• For if it be poflible, let one of them -i greater than the other ; be In X \ and X ^ we have M = A (pr.4.) BOOK I. PROP. XXVI. THEOR. 27 but A = iH (hyp.) and therefore g^^ =: ^|B, which is abfurd ; hence neither of the fides — ^— ■— and — ■^■■■- is greater than the other; and .*. they are equal; and 4 = 4, (pr. 4.). CASE II. Again, let ^— — • ^ ■— — — ^ which lie oppofite the equal angles flik and ^^^ . If it be poflible, let Then in ' ^ and J^^^ we have = and /^ = J^, I'ut H^ = JBi^ (hyp.) .*. jf^ = ^^^ which is abfurd (pr. 16.). Confequently, neither of the fides ^"i— i"«» or ^-^"i—^ is greater than the other, hence they muft be equal. It follows (by pr. 4.) that the triangles are equal in all refpedls. Q^E. D. 28 BOOK I. PROP. XXVII. THEOR. F ajlralght line ( ) meet- i?2g tivo other Jiraight lines, - and ) makes with them the alternate angles ( and ) equal, thefe two Jiraight lines are parallel. If be not parallel to they fliall meet when produced. If it be poflible, let thofe lines be not parallel, but meet when produced ; then the external angle ^^ is greater than flHik>^ (pr. i6),but they are alfo equal (hyp.), which is abfurd : in the fame manner it may be ihown that they cannot meet on the other fide ; ,*, they are parallel. Q. E. D. BOOK I. PROP. XXFIIL THEOR. 29 (- F ajlraight line ting two other Jlraight lines makes the external equal to the internal and oppojite angle, at the fame Jide of the cutting line {namely. yl, or if it makes the two internal angles at the fame ftde ( ^l^ and ^F , or f/^ and ^^^) together equal to two right angles, thofe two Jlraight lines are parallel. Firft, if 1^ =^^ , then Jjj^ = ^r (pr. i mL = W /. II (pr. 27.). Secondly, if then + (pr. 13.), (ax. 3.) (pr. 27.) C^E. D. 30 BOOK I. PROP. XXIX. THEOR. STRAIGHT /ine ( ) f^^^i'"g on two parallel Jiraight » lines ( ■mmmim^mm and •), makes the alternate angles equal to one another ; and alfo tlie external equal to tlie in- ternal and oppojite angle on the fame Jide ; and the two internal angles on the fa?ne Jide together equal to two right angles. For if the alternate angles and ▲ be not equal, draw », making A Therefore (pr- 23)- (pr. 27.) and there- fore two ftraight lines which interfed: are parallel to the fame flraight line, which is impoflible (ax. 1 2). Hence the alternate angles ^^ and ^|^ are not unequal, that is, they are equal: =: ^^^ (pr. 15); .*. jl^ = l/^ , the external angle equal to the inter- nal and oppofite on the fame iide : if ^^W be added to both, then A + i ^CLi (pr. 13)- That is to fay, the two internal angles at the fame fide of the cutting line are equal to two right angles. Q. E. D. BOOK I. PROP. XXX. THEOR. 3^ TRAIGHT /mes ( _Z) lohich are parallel to the fame Jlratght line ( ), are parallel to one another. Let interfedl Then, = ^^ = iJB (pr. 29.), (pr. 27.) Q. E. D. 32 BOOK I. PROP. XXXI. PROB. ROM a given point /^ to draw ajiraight line parallel to a given Jlraight line (——•). Draw from the point / to any point in make then — (pr. 23.), - (pr. 27.). Q. E. D. 4 BOOK I. PROP. XXXII. THEOR. 33 F any fide (- •) of a triangle be pro- duced, the external ^figl^ ( ^^^) '-^ ^qual to the fum of the two internal and oppofte angles ( aiid ^^^ ) , and the three internal angles of every triangle taken together are equal to two right angles. Through the point / draw II (pr. 3i-)- Then (pr. 29.), and therefore (pr. 13.). J -dy Q. E. D. 34 BOOK I. PROP. XXXIII. THEOR. TRAIGHT fines (- and ) which join the adjacent extremities of two equal and parallel Jiraight ~— — and "•»..---=. ), are themf elves equal and parallel. Draw the diagonal. (hyp.) and (pr. 29.) common to the two triangles ; ■, and ▼ = 4 (pr. 4.) ; and /. (pr. 27.). Q. E. D. BOOK I. PROP. XXXIV. THEOR. 35 HE ofpofite Jides and angles of any parallelogram are equal, and the diagonal (i^— ^^— ) divides it into two equal parts. Since = A ^ = t (pr. 29.) and common to the two triangles. /. \ \ (pr- 26.) and ^^W = ^^M (^^'^ ' Therefore the oppofite fides and angles of the parallelo- gram are equal : and as the triangles .N.""^ are equal in every refpect (pr. 4,), the diagonal divides the parallelogram into two equal parts. Q. E. D. 36 BOOK I. PROP. XXXV. THEOR. ARALLELOGRAMS on the fame bafe, and between the fame paral- lels, are {in area) equal. and But, On account of the parallels, _Kpr. 29.) (Pi-- 34-) (pr. 8.) r=? minus minus r= Q^E. D. BOOK I. PROP. XXXVI. THEOR. 37 ARALLELO- GRAMS 1 is* ( ^^ and ) on equal bafes, and between the fame parallels, are equal. Draw and ---..-— , ■, by (pr. 34, and hyp.); = and II "— (pr. 33.) And therefore but J 1.1 is a parallelogram : (pr- 35-) (ax. I.). Q. E. D. 38 BOOK I. PROP. XXXFII. THEOR. RIANGLES on the fame bafe (• ■) and between the fame paral- lels are equal. Draw Produce \ fpr. ^i (pr- 3I-) 1—M. and ^^ are parallelograms on the fame bafe, and between the fame parallels, and therefore equal, (pr. 35.) T =: twice f ^ twice 4 (■ (pr- 34-) k.i Q. E D. BOOK I. PROP. XXXVIII. THEOR. 39 RIANGLES ;4H ^'ij JH (^Hi tind jm^ ) on equal bajes and between •■• the fame parallels are equal. Draw and II (pr. 31.). I # (pr. 36.); and ■ i = twice ^^k ^^ = twice ^H (pr- 34-)' i k (ax. 7.). Q^E. D. 40 BOOK I. PROP. XXXIX. THEOR, QUAL triangles \ and "^ on the fame bafe ( ) and on the fame fide of it, are between the fvne parallels. If-^— ■», which joins the vertices of the triangles, be not || , draw II (pr.3i-). meeting Draw Becaufe (conft.) but W.4 (pr- 37-) ■• (hyp.) ; A=4 , a part equal to the whole, which is abfurd. Ji. ^i^-^-^ ; and in the fame manner it can be demonflrated, that no other line except is II ; .-. II . Q. E. D. BOOK I. PROP. XL. THEOR. 41 QUAL trian- gles ( and M. ) on equal bafes, and on the fame Jide, are between the fame parallels. If ■ which joins the vertices of triangles be not II - , draw — — — . II — -~— (pr. 31.), meeting Draw Becaufe (conft.) . ^^^- ^ 1^^^ , a part equal to the whole, which is abfurd. ' 41" ~^^^"^ • ^"f^ in the fame manner it can be demonftrated, that no other line except — is II : .-. Q^E. D. 42 BOOK I. PROP. XLI. THEOR. Draw Then F a paral- lelogram A and a triangle ^^^ are upon the fame bafe — ^^^ and be tine en the fame parallels -.—---- and ■ , the parallelogram is double the triangle. the diagonal ; V=J zz twice (pr- 37-) (pr- 34-) ^^ 4 ^1^. ^ twice ^H^ . Q. E. D. BOOK I. PROP. XLII. THEOR. 43 O conJiruSl a parallelogram equal to a given 4 triangle ^^/^andhaV" ing an angle equal to a given rectilinear angle , Make — — ^ = ■— « (pr. 10.) Draw , Draw I " [j ~' | (pr. 31.) ^1^ := twice y (pr. 41.) but ^ z= lA (pr. 38.) 4 Q. E. D. 44 BOOK I. PROP. XLIII. THEOR. HE complements and ^^^ cf the parallelograms ivhicli are about the diagonal of a parallelogram are equal. (pr- 34-) 4. ^^ and JBL = ^ (pi-- 34-) (ax. 3.) Q. E. D. BOOK I. PROP. XLIV. PROB. 45 O a given Jlraight line ply a parallelo- gram equal to a given tri- angle ( ^^^^' ), and having an angle equal to a given reSiilinear angle ( )■ g wi th ▲ = ._i Make (pr. 42.) and having one of its fides -— — - conterminous with and in continuation of 1 m . Produce w^^mmm^ till it meets ' '■ " ■' || »»»■«»■ draw prnHnrp it fill if mpptg •■»■-,• continued ; draw •••««-.• II — «■■ meeting produced, and produce >•»■•»« but (pr. 430 (conft.) ▲ = ▼=▲ (pr. 19. and confl.) Q. E. D. BOOK I. PROP. XLF. PROP. O conjlrudl a parallelogram equal to a given reSlilinear figure ( ► ) and having an angle equal to a given reSlilinear angle Draw and K.(t^m dividing to the redtilinear figure into triangles. Conftrudl having .„ — apply (pr.42.) having to having (pr. 44.) apply M = (pr. 44.) #=► ##= >, and Mf mg is a parallelogram, (prs. 29, 14, 30.) having Q. E. D. BOOK I. PROP. XLVI. PROB. 47 PON a given Jlraight line (— ^^^) to conJlruB a fquare. Draw Draw • ing . and (pr. 1 1, and 3.) II drawn || >, and meet- In ^ (conft.) =: a right angle (conft.) ^H = Hp = ^ "g'^^ ^"gle (pr. 29.), and the remaining fides and angles muft be equal, (pr. 34.) and ,*, is a fquare. (def. 27.) Q. E. D. 48 BOOK I. PROP. XLVII. THEOR. N a right angled triangle the fquare on the liypotenufe <• •< is equal to the fum of the fquares ofthejides, (■ and ). On and defcribe fquares, (pr. 46.) Draw -.—I alfo draw - (pr. 31-) and To each add T and Again, becaufe BOOK I. PROP. XLVII. THEOR. 49 and twice = twice ^H • In the fame manner it may be fhown that ^^ ^ hence ## Q E. D. H so BOOK I. PROP. XLVIIL THEOR. / F t/ie fquare of one Jide { \ ) f a triangle is equal to the fquares of the other tivo fides (nn. i i and ), the angle ( )fubtended by that fide is a right angle. Draw ■- and ^ (prs.11.3.) and draw —»-«--— alfo. Since (conft.) ... "- + but ^ + - and — ^— i^- -|- + (pr. 47-). - (hyp.) and ,*, confequently (pr. 8.), is a right angle. Q. E. D. 51 BOOK II. DEFINITION I. RECTANGLE or a right angled parallelo- gram is faid to be con- tained by any two of its adjacent or conterminous fides. Thus : the right angled parallelogram HH[ be contained by the fides — — — ^ and — or it may be briefly defignated by is faid to If the adjacent fides are eq^ual ; i. e. -— — — ^ ^ then — i^»^-« . - which is the expreflion for the redtangle under is a fquare, and is equal to J and - or - or 52 BOOK II. DEFINITIONS. DEFINITION II. N a parallelogram, the figure compokd of one ot the paral- lelograms about the diagonal, together with the two comple- ments, is called a Gnomon. Thus and are called Gnomons. BOOK II. PROP. I. PROP. 53 HE 7-e£langle contained by two ftraight lines, one of which is divided into any number of parts. = <;+ — /; equal to the fum of the reBangks contained by the undivided line, and the fever al parts of the divided line. I — — J— — i; Draw _L —— — and r= (prs.2.3.B.i.); complete the parallelograms, that is to fay, Draw \ (pr. 31- B.I.) L I + - + Q. E. D. 54 BOOK II. PROP. II. THEOR. I I F a Jlraight line be divided into any tivo parts ' i , the fquare of the -whole line is equal to the fum of the reSlangles contained by the whole line and each of its parts. -f I Defcribe ■■-^^ (B. i. pr. 46.) Draw — parallel to ----- (B. i. pr. 31 ) I + Q. E. D. BOOK 11. PROP. III. THEOR. 55 F a Jiraig/it line be di- vided into any two parts ■ 11 ' , the reBangle contained by the "whole line and either of its parts, is equal to the fquare of that part, together with the reSf angle under the parts. m i = — ^ + or. Defcribe Complete I (pr. 46, B. I.) (pr. 31, B. I.) Then + , but and In a fimilar manner it may be readily fhown that — . — zr m^'i _^ ——. — . Q. E. D 56 BOOK II. PROP. IF. THEOR. F a Jiraight line be divided into any tico parts , the fquare of the ii'hole line is equal to the fquare s of the parts, together ii-ith twice the reef angle contained by the parts. + + twice Defcribe draw - and 4- vpr. 46, B. 1.) ■ port. I.). (pr. 31, B. I.) 4.4 (pr. 5, B. I.), (pr. 29, B. I.) 4 500a: //. PROP. IF. THEOR. 57 B /. by (prs.6,29, 34. B. I.) t,^J is a fquarc ^ — i For the fame reafons r I is a Iquare := ~"", « ""~ (pr, 43, b. I.) I b"t E— i = C-J+ — +— + B. twice >' • ■— , Q. E. D. 58 BOOK 11. PROP. V. PROP. F a Jlraight line be divided into two equal parts and alfo ^ into two unequal parts, the reSlangle contained by the unequal parts, together with the fquare of the line between the points of fe 51 ion, is equal to the fquare of half that line + Defcribe IIHIH (pr. 46, B. i.), draw ^ — II — -- and ) II (pr.3i,B.i.) (p. 36, B. I.) ■ - H (p. 43. B. I.) (ax. 2.) I- BOOK II. PROP. r. THEOR. 59 but and - (cor. pr. 4. B. 2.) (conft.) /. (ax. 2.) ifl.F- + Q. E. D. 6o BOOK II. PROP. VI. THEOR. F a Jlraight line be bifeSled ■ and produced to any point —^wmmmt , the reSlangle contained by the whole line fo increafed, and the part produced, together with the fquare of half the line, is equal to the fquare of the line made up of the half, and the produced part . Defcribe (pr. 46, B. I.), draw II and (pr. 3i,B. 1.) (prs. 36, 43, B. I ) but ^H = (cor. 4, B. 2.) + (conft.ax.2.) Q. E. D. BOOK 11. PROP. VII. THEOR. F a Jlraight line be divided into any two parts wbmw^— , the fq liar es of the whole line and one of the farts are equal to twice the rectangle contained by the whole line and that part, together •with the fquare of the other parts. 6i Defcribe Draw - and ■ ■^■■■■« , (pr. 46, B. I.)- (poft. I.), (pr. 31, B. !.)• — I (pr- 43. -B. I.), add ■ = ■-' to both, (cor. 4, B- 2.) I (cor. 4, B. 2.) I + ■ + + ■' + — ^ = + Q. E. D. 62 BOOK II. PROP. VIII. THEOR. E3 F ajlraight line be divided Into any two parts , the fquare of thefum of the whole line and any one of Its parts. Is equal to four times the reSlangle contained by the whole line, and that part together with the fquare of the other part. — + Produce and make Conftrudl draw J (pr. 46, B. 1.); (pr. 31, B. I.) but ^ + (pr. 4, B. II.) -^ z= 2. — (pr. 7, B. II .) •-+ — ^ + °-' Q. E. D. BOOK 11. PROP. IX. THEOR. F a Jlraight line be divided into two equal parts ^— — ,.j y 63 and alfo into two unequal parts ^mmm^'^^m— ^ the fquares of the unequal parts are together double the fquares of half the line, and of the part between the points offedlion. ^ + ^= 2 ^ + 2 Make — ■ _L and r= — Draw "..-.—«— and — II ,— II or and draw = 4 4. = ^ (pr. 5, B.I.) ^ half a right angle, (cor. pr. 32, B. i.) (pr. 5, B. I.) =: half a right angle, (cor. pr. 32, B. i.) ^ a right angle. 4^ lence (prs. 5, 29, B. I.). wmmimtm^m^ ■■■■■* (prs. 6, 34, B. I.) + ^or + I I ■ + \ (pr. 47, B. I.) + 2 Q. E. D. 64 BOOK II. PROP. X. THEOR. F a Jlraight line ■ be bi- feBed and pro- duced to any point • — , thefquaresofthe •whole produced line, and of the produced part, are toge- ther double of the fquares of the half litie, and of the line made up of the half and pro- duced part. + + ^ Make and ■— J_ and =1 to draw ^MvatMit and or - f (pr. 31, B. I.); draw alfo. 4 (pr. 5, B. I.) = half a right angle, (cor. pr. 32, B. i .) (pr. 5, B. I.) = half a right angle (cor. pr. 32, B. i.) 4. m a right angle. BOOK II. PROP. X. THEOR. 6^ half a right angle (prs. 5, 32, 29, 34, B. i.), and .-.., (prs. 6, 34, B. I.). Hence by (pr. 47, B. i.) Q. E. D. 66 BOOK II. PROP. XI. PROP. O divide a given fir aight line -^^■■» in fuch a manner, that the reB angle contained by the whole line and one of its parts may be equal to the fquare of the other. Defcribe make «««■ 1 1 ■ • «*»■ a n draw take on defcribe (pr. 46, B, I.), - (pr. 10, B. I.), (pr. 3, B. I.), (pr. 46, B. I.), Produce — (poft. 2.). Then, (pr. 6, B. 2.) 2 _ i + • •■■••■■ ■"■', or, I Q^E. D. BOOK II. PROP. XII THEOR. 67 N any obtufe angled triangle, thefquare of the fide fubtend- ing the obtufe angle exceeds the fiim of the fquares of the fides containing the ob- tufe angle, by twice the rec- tangle contained by either of thefe fides andthe produced parts of the fa?ne from the obtufe angle to the perpendicular let fall on it from the oppofite acute angle. + '' by ^ + 2 • + By pr. 4, B. 2. ^ + > + 2 add — — — ^ to both 2 _ V (pr. 47, B. I.) + + • or ■ ; hence ' by 2 '^ (pr. 47, B. I.). Therefore, ^ • ' -"■ + ' + Q. E. D. 68 BOOK II. PROP. XIII. THEOR. FIRST. SECOND. ^m F^ p^ Br^/^ ^ N any tri- angle, the fquareofthe Jidefubtend- ing an acute angle, is lefs than the fum of the fquares of the Jides con- taining that angle, by twice the reSlangle contained by either of thefe fides, and the part of it intercepted between the foot of the perpendicular let fall on it from the oppofte angle, and the angular point of the acute angle. FIRST. + ■ * by 2 SECOND. .' -I *by 2 + 2 • Firft, fuppofe the perpendicular to fall within the triangle, then (pr. 7, B. 2.) ^■■■> ° -|- ^^^— ^ ^ 2 • ^^i^^"»« • — — -^ ■■■• add to each ^ihi^'^ then, I..... "■ -|- _• ^4- - = 2 • ■— • - + ' + « /. (pr- 47. B. I.) + BOOK 11. PROP. XIII. THEOR. 69 and .*. ^ Z] ^— "— - + — - by 2 • -■" • ■■■-■i™ . Next fuppofe the perpendicular to fall without the triangle, then (pr. 7, B. 2.) add to each — ■— ■ - then + ^ + 2 ... (pr. 47, B. I.), ■J 1 <2 ^_ „ I a 1^— -|- -^.— ^ 2 • ^mM»» . _l-_ -J- ', Q. E. D. 7° BOO A' //. PROP. XIV. PROB. O draw a right line of •which the fquare flmll be equal to a given reSli- linear figure . fuch that. Make ^^^^H = ^^V (pr. 45, B. i.), produce "•- until — — -■. := • take -■■■.«—- ^ i^— — (pr. 10, B. i.), Defcribe f \ (poft. 3.), and produce -^^— to meet it : draw — — ^— , (pr. 5, B. 2.), but — ■ = ' ' " + — "— -(pr. 47, B. I.); • wmmm^t" ^I ■■■■■■ • «■»« , and Q. E. D. BOOK III. DEFINITIONS. I. QUAL circles are thofe whofe diameters are equal. II. A right line is said to touch a circle when it meets the circle, and being produced does not cut it. III. Circles are faid to touch one an- other which meet but do not cut one another. IV. Right lines are faid to be equally diftant from the centre of a circle when the perpendiculars drawn to them from the centre are equal. 72 DEFINITIONS. And the ftraight line on which the greater perpendi- cular falls is faid to be farther from the centre. VI. A fegment of a circle is the figure contained by a ftraight line and the part of the circum- ference it cuts off. VII. An angle in a fegment is the angle con- tained by two ftraight lines drawn from any point in the circumference of the fegment to the extremities of the ftraight line which is the bafe of the fegment. VIII. An angle is faid to ftand on the part of ; the circumference, or the arch, intercepted between the right lines that contain the angle. IX. A fed:or of a circle is the figure contained by two radii and the arch between them. DEFINITIONS. 11 Similar fegments of circles are thofe which contain equal angles. Circles which have the fame centre are called concentric circles. 74 BOOK III. PROP. I. PROB. O Jind the centre of a given circle o Draw within the circle any ftraight Hne — ^ draw hi left . ma ke. i^MMMi ^ and the point of biledtion is the centre. For, if it be pofTible, let any other point as the point of concourfe of .^— — , ---..--- and — .— — be the centre. Becaufe in and ■ ----— (J'^yp- ^""^ 2* I J def. 15.) -- (conft.) and ••■- common, ^B. I, pr. 8.), and are therefore right angles ; but ^ = ^_| (con ft. (ax. I I .) which is abfurd ; therefore the aflumed point is not the centre of the circle ; and in the fame manner it can be proved that no other point which is not on — ^^— • is the centre, therefore the centre is in ^— ^^— , and therefore the point where 1 is bifedled is the centre. Q. E. D. BOOK III. PROP. 11. THEOR. 75 STRAIGHT line C—) joining two points in the circumference of a circle lies ivholly within the circle. Find the centre of o (B.S-pr.i.); from the centre draw to any point in meeting the circumference from the centre ; draw — — — and . Then = -^ (B. i.pr. 5.) but or CZ ^ (B. I. pr. 16.) (B. I. pr. 19.) but .*. every point in lies within the circle. Q. E. D. 76 BOOK III. PROP. III. THEOR. Draw F a Jlraight line ( drawn through the centre of a circle o bife£lsachord ( •'•■) which does not paj's through the centre, it is perpendicular to it; or, if perpendicular to it, it bifeSls it. and to the centre of the circle. In >^ I and L..._V ■• •■ ■■» and ,*, m^^^ common, and = (B. 1. pr. 8.) _L -..«.- (B. I. def. 7.) Again let Then in J and L^„..T^ (B. i.pr. 5.) (hyp.) and and .*. (B. I. pr. 26.) bifedts Q. E. D. BOOK HI. PROP. IF. THEOR. 11 F in a circle tiaojlraight lines cut one another, which do not both pafs through the centre, they do not bifeSl one another. If one of the lines pafs through the centre, it is evident that it cannot be bifecfted by the other, which does not pafs through the centre. But if neither of the Hnes — =— ^^— or •— ^-i— pafs through the centre, draw ——----. from the centre to their interfedlion. If «i^^^ be bileded, ._._._ _L to it (6. 3. pr. 3.) .*. ^^ = I ^ and if — be bifed:ed, (B. 3- P'-- 3-) and .*, 5 a part equal to the whole, which is abfurd : .*. — —— — and — — — • do not biiecfl one another. Q. E. D. w* 78 BOOK III. PROP. V. THEOR. F two circles interfeSl, they have not the (0) Janie centre. Suppofe it poflible that two interfedting circles have a common centre ; from fiich fuppofed centre draw ^.i^.. to the interfering point, and ^^—^— ••■--■■ the circumferences of the circles. meetmg Then and <— (B. i.def 15.) - (B. I. def. 15.) «»- J a part equal to the whole, which is abfurd : .', circles fuppofed to interfedl in any point cannot have the fame centre. Q,E. D. BOOK III. PROP. VL THEOR. 79 F tivo circles touch one another internally, they have not the fame ce?itre. For, if it be poffible, let both circles have the fame centre; from fuch a fuppofed centre draw ---■» cutting both circles, and ■— — ^— to the point of contadl. Then and — (B. i.def. 15.) (B. I.def. 15.) J a part equal to the whole, which is abfurd ; therefore the aiTumed point is not the centre of both cir- cles ; and in the fame manner it can be demonftrated that no other point is. g E. D. 8o BOOK HI. PROP. FII. THEOR. nCURE 1. FIGURE II. F Jt'om any point within a circle which is not the centre, lines o are drawn to the circumference ; the greatejl of thofe lines is that (-i^.«i"») which pajfes through the centre, and the leaf is the remaining part ( ^ of the diameter. Of the others, that ( ^— ■— — > ) which is nearer to the line pafing through the centre, is greater than that ( mmmmm^^ ) wliich Is itiore remote. Fig. 2. The two lines (• and ) which make equal angles with that pafpng through the centre, on oppofite fdes of it, are equal to each other; and there cannot be drawn a third line equal to them, from the fame point to the circumference. FIGURE I. To the centre of the circle draw —-—— and -— «-- — j then "— -— rr —..—.. (B. i. def. 15.) ......i^Mi ^ ^^— -|- ...«>. C — — ^-» (B.I . pr. 20.) in like manner — — — (• may be fhewn to be greater than .i.M__- , or any other line drawn from the fame point to the circumference. Again, by (B. i. pr. 20.) take — — from both ; /. — — — CI ....1^— (ax.), and in like manner it may be fhewn that — ^— ^ is lefs BOOK III. PROP. VII. THEOR. 8i thiin any other line drawn from the fame point to the cir- cumference. Again, in y*/ and common, ^^ [^ IV , and (B. I. pr. 24.) and may in like manner be proved greater than any other line drawn from the fame point to the circumference more remote from -——■-, FIGURE II. If ^=^. hen = ,if„o. take — — ^— r= ^— — draw '■"■'■, then ^c— 'I :>»^ in ^^ I and I ,^^ , — — common. (B. i.pr. 4.) a part equal to the whole, which is abfurd : ■"— — IS * and no other line is equal to ■^ drawn from the fame point to the circumfer- ence ; for if it were nearer to the one pafling through the centre it would be greater, and if it were more remote it would be lefs. Q. E. D. M 82 nOOK HI. PROP. nil. THEOR. The original text of this propolition is here divided into three parts. I. ^Voll ^^ f''°'" '' P°'"^ without a circle, Jlraight iiKCs \ — — — \ ore (jrd'wn to V»y the cir- cu/nference ; of thofe falling upon the concave circum- ference the greatejl is that ( — ») ichich fafja through the centre, and the line ( i ) ichich is nearer the greatejl is greater than that ( ) 'ichich is more remote. Draw and .■■-..>... to the centre. Then. ■— . which pallcs through the centre, is greatcll; for fince — — nz , it ^-^-^— be added to both. -•■-■ ^ — ^— -|- ; l^iit C — — — {^- 1- P'- -^•) •'- — — — is greater than ;inv other line dr.iwn from the fame point to the concave circumference. Again in and BOOK rrr. prop. nir. tiikor and — ^— ^ common, Init (B. I. pr. 24.); and in like manner may be Ihcwn ZZ t'l-"! -^'7 other line more remote from II. Of thofc lines falling on the convex circumference the leaf is that (————) which being produced would pafs through the centre, and the line which is nearer to the lea/l is Icf than that which is more remote. For, fince — — — -j- and /« And again, fince — — -|- h (B. i.pr. 21.), and — — rs . — — -, And lb of others. III. Alfo the lines making equal angles with that which pajjes through the centre arc equal, whether /ailing on the concave or convex circumfrence ; and no third line can he drinvn equal to the/n from the f<imc point to the circumference. Forif •■ make •— ^ C "■"•"", Init making ^ =: ^ ; ----- ^ ----- ^ ;iml tl|;l\V ...... -.^ 84 BOOK III. PROP. Fin. THEOR. Then in and ) and / ; / we have common, and alfo ^ =: 41, - = (B. I. pr. 4.); but which is abfurd. of ■ ■■iisBisB IS not Z!Z ----- — 9 .*. --■■ -, nor to any part is not r~ — -----^ Neither is '— , they are to each other. And any other line drawn from the fame point to the circumference muft He at the fame fide with one of thefe lines, and be more or lefs remote than it from the line pall- ing through the centre, and cannot therefore be equal to it. Q. E. D. BOOK in. PROP. IX. THEOR. 85 F a point be taken ivithin a circle ( ] , from which o- more thwi two equal Jlraight lines can be drawn to the circumference^ that point tnuji be the ceiitre of the circle. For, if it be fuppofed that the point |^ in which more than two equal ftraight lines meet is not the centre, fome other point — .. muft be; join thefe two points by and produce it both ways to the circumference. Then fince more than two equal ftraight lines are drawn from a point which is not the centre, to the circumference, two of them at leall; muft lie at the fame fide of the diameter .; and fince from a point /\ w hich is not the centre, ftraight lines are drawn to the circumference ; the greateft is -i^— .--= = ^ which pafies through the centre : and — ^— — — which is nearer to ^^■—••«'. |^ ———~- which is more remote (B. 3. pr. 8.) ; but — — — ^— rr ^— ^— =- (hyp.) which is abfurd. The fame may be demonftrated of any other point, dif- ferent from / |\^ which muft be the centre of the circle, Q. E. D. 86 BOOK III. PROP. X. THEOR. NE circle I } cannot interfeSl another I J /« more points than two. For, if it be poffible, let it interfedl in three points ; from the centre of ( I draw O to the points of interfedlion ; (B. I. def. 15.), but as the circles interfedl, they have not the fame centre (B. 3. pr. 5.) : .*, the alTumed point is not the centre of o. and and are drawn from a point not the centre, they are not equal (B. 3. prs. 7, 8) ; but it was fhewn before that they were equal, which is abfurd ; the circles therefore do not interfedl in three points. Q. E. D. BOOK III. PROP. XL THEOR. 87 F two circles o o and touch one another internally, the right line joining their centres, being produced, Jliall pafs through a point of contaSl. For, if it be poffible, let join their centres, and produce it both ways ; from a point of contadl draw — — — to the centre of ( J , and from the fame point of contadl draw -•-•--— to the centre of I J . Becaufe in 4 + ■■■•■•■•t (B. I . pr. 20.), and o as they are radii of 8B BOOK III. PROP. XI. THEOR. but — ^ -|- I rr — — — ; take away — — — which is common, hut — i^— i ^ -- — --^ ii of r^ , becaufe they are radi and ,*, --»-" C ^^ ^ P^i't greater than the whole, which is abfurd. Tlie centres are not therefore fo placed, that a line joining them can pafs through any point but a point of contadt. Q. E. D. BOOK III. PROP. XII. THEOR. 89 F two circles o t/ier externally, the Jiraight line 1 1 joining their centres, pajfes through the point of contaB. touch one a7io If it be polTible, let join the centres, and not pafs through a point of contadl ; then from a point of contad: draw -"^^^== and "" ■ ••^■^- ' '-• to the centres. Becaufe and . and • . + (B. I. pr. 20.), = (B. I. def. 15.), = ^ (B. I. def. 15.), + ', a part greater than the whole, which is abfurd. The centres are not therefore fo placed, that <-he line joining them can pafs through any point but the point of contadl. Q.E. D. 90 BOOK in. PROP. XIIL THEOR. FIGURE I. FIGURE II. FIGURE III. NE circ/e can- not touch ano- ther, either externally or internally, in more points than one. Fig. I . For, if it be poffible, let and f 1 touch one o another internally in two points ; draw — — - joining their cen- tres, and produce it until it pafs through one of the points of contadl (B. 3. pr. 11.); draw — — — and •^— ^—^ , .-. if (B. I. def 15.), be added to both, + but and .*. + + which is abfurd. .■ (B. I. def 15.), - = — _— ; but — — (B. I. pr. 20.), BOOK III. PROP. XIII. THEOR. 91 Fig. 2. But if the points of contadl be the extremities of the right line joining the centres, this ftraight line muft be bifedled in two different points for the two centres; be- caufe it is the diameter of both circles, which is abfurd. , let f j and I J Fig. 3. Next, if it be poffible touch externally in two points; draw — — joining the centres of the circles, and paffing through one of the points of contact, and draw — i— — — and -^^— — , — -^ z= _ (B. I. def. 15.); nd ...«■■•. zr I (B. i. def. 15.); -\- — — -^ Z:Z. ■BMBsaaa * but + ^-^™''— C ».-- (B. I. pr. 20.), which is abfurd. There is therefore no cafe in which two circles can touch one another in two points. Q E. D. 92 BOOK III. PROP. XIV. THEOR. Then and infcribed in a circle are e- qually dijiantfrom the centre ; andalfofjlraight lines equally dijiafit from the centre are equal. From the centre o o draw -L to — ,join ■••■■ and --• ^— and — ■ fince = half '" (B. 3. pr. 3.) = i (B. 3-pr-3-) = — (hyp.) and (B. I. def. 15.) and but fince > - s^ is a right angle = ' + MB.i.pr.47-) .' = ' + ^ for the .% -^ + fame reafon. BOOK III. PROP. XIV. THEOR. 93 Alfo, if the lines 1 . .■.•»■ and •— i»«...«.r be equally diflant from the centre ; that is to fay, if the per- pendiculars -■-■•■■■■■■ and -m........ be given equal, then For, as in the preceding cafe, . ^ :::= __i.^, and the doubles of thefe ....... and ^n....... are alfo equal. Q. E. D. 94 BOOK III. PROP. XV. THEOR. FIGURE I. but HE diameter is the greatejl Jiraight line in a circle : and, of all others, that which is neareji to the centre is greater than the more remote. FIGURE I. The diameter ^^ is CZ any line For draw ' and •••••••••• Then .••■>••»■■> ^ ^— ^._i and •^— ^— = — — — . mXm ■■■■•»««» ^^ ■■■•■■•■• (B. I . pr. 20.) Again, the Hne which is nearer the centre is greater than the one more remote. Firft, let the given lines be — — ^ and - , wnich are at the fame fide of the centre and do not interfed: ; draw J ' BOOK III. PROP. XF. THEOR. 95 FIGURE II. Let the given lines be ■^— ^ and ^i— ^ which either are at different fides of the centre, orinterfedt; from the centre draw - -■•-- and -»-«-—- J_ -^^Mi->i» and , FIGURE II. make draw Since and the centre, but — — — and are equally diflant from (B. 3. pr. 14.); (Pt. i.B. 3.pr. 15.), Q. E. D. 96 500 A' ///. PROP. XVI. THEOR. llEJiraiglit line - draii-n from the extremity of the diame- ter of a circle h perpendicular to it falls ••.^^ ^,., without the circle. ||» • '^ * And if anyjlraight *** line ........ be drawn from a point ————— within that perpendi- cular to the point of contaB, it cuts the circle. PART I If it be poffible, let ^ which meets the circle again, be _L , and draw , Then, becaufe ^ = ^ (B. i.pr. 5.), and .'. each of these angles is acute. (B. i. pr. 17.) but ^^ =r I J (hyp.), which is abfurd, therefore ii...._ drawn _L — ^^^— does not meet the circle again. BOOK in. PROP. XVI. THEOR. 97 PART 11. Let — Bi*—"— be _L -^-^^— and let -— — - be drawn from a point y between — ■— ■— • and the circle, which, if it be poflible, does not cut the circle. Becaufe H^ =: | ^ , .*. ^^ is an acute angle ; fuppofe ....... ...4.... _L .-■«•-•-, drawn from the centre of the circle, it mufl; fall at the fide of ^^ the acute angle. ,*, B^^ which is fuppofed to be a right angle, is C ^^; but ............ = , and .*, --••••.. ^ ......«..■—, a part greater than the whole, which is abfurd. Therefore the point does not fail outfide the circle, and therefore the ftraight line ■ ••••MiM* cuts the circle. Q. E. D. 98 BOOK III. PROP. XVII. THEOR. O Jraiv a tangent to a given circle \ \ from a o given point, either in or outjide of its '•♦^ circumference. If the given point be in the cir- cumference, as at I , it is plain that the ftraight line "■■" J_ -«— — - the radius, will be the required tan- gent (B. 3. pr. 16.) But if the given point ^ outlide of the circumference, draw — from it to the centre, cutting be draw •■««■■■■** ^_ concentric with then o .., defcribe radius^ .■■■■ub^, will be the tangent required. BOOK III. PROP. XFII. THEOR. 99 XV / / For in and i\. , ^^^ common, (B. I. pr. 4.) flB =: ^^^ ^ a right angle, .*. ^a^a^B is a tangent to Q. E. D. o loo BOOK III. PROP. XFIII. THEOR. and .*, F a right line •• — be a tangent to a circle, the fir aight line — ^— draivn from the centre to the i point of contaSl, is perpendicular to it. For, if it be poflible, let *>■ be ^ — ... then becaufe = [^ is acute (B. i . pr. 17.) c (B. I. pr. 19.); but »•*•■ , a part greater than the whole, which is ablurd. /, .»«.. is not _L -"•—•••5 and in the fame man- ner it can be demonftrated, that no other line except — ■— — is perpendicular to «•-.■...-• , Q. E. D. BOOK III PROP. XIX. THEOR. lOI F a Jlra'tght line be a tangent to a circle, thejiraight line , drawn perpendicular to it from point of the contact, pajfes through the centre of the circle. For, if it be pofTible, let the centre be without « and draw • ••- from the fuppofed centre to the point of contadl. Becaufe X (B. 3.pr. i8.) .'. ^^ =: I Ji , a right angle ; but ffj^ = I 1 (hyp.)' and /. ^ = a part equal to the whole, which is abfurd. Therefore the affumed point is not the centre ; and in the fame manner it can be demonftrated, that no other point without _„_^ is the centre. Q. E. D. 102 BOOK III. PROP. XX. rUEOR. FIGURE I HE angle at the centre of a circle, is double the angle at the circumference, when they have the fame part of the circumference for their bafe. FIGURE I. Let the centre of the circle be on a fide of ^ , Becaufe i = ^ = ^ (B. i.pr.5.). But + \ or =: twice . (B. i. pr. 32). FIGURE II. FIGURE II. Let the centre be within ^ ^ the angle at the circumference ; draw ■■■^^^— from the angular point through the centre of the circle ; then ^ := r ? and = ^^ , becaufe of the equality of the fides (B. i. pr. 5). BOOK III. PROP. XX. THEOR. 103 Hence -|- ^ -|~ "I" ^ twice But ^ = ^ + ^ , and r= twice FIGURE III. Let the centre be without W and FIGURE III. draw m^ the diameter. B( jcaufe ▼ : twice ^ ; i r= twice ^^ (cafe I .) ; • • • A ^ twice ▼ , Q. E. D. I04 BOOK III. PROP. XXI. THEOR. FIGURE I. HE angles ( ^^ , ^^ ) in the fame Jegment of a circle are equal. FIGURE I. Let the fegment be greater than a femicircle, and draw — ^^^— and — — — — to the centre. twice ^^ or twice (B. 3. pr. 20.),- 4 = 4 4 FIGURE II. FIGURE II. Let the fegment be a femicircle, 01 lefs than a femicircle, draw —— — the diameter, alfo draw ^=4a„dV = ^ (cafe I.) Q. E. D. J BOOK III. PROP. XXII. THEOR. 105 HE oppojite arigJes Af and ^^ . ^^1 and ^r of any quadrilateral figure in- fcr'ibed in a circle, are together equal to two right angles. Draw and the diagonals ; and becaufe angles in the fame fegment are equal ^W =: ^^ and ^r rr: ^^ | add ^ to both. two right angles (B. i. pr. 32.). In like manner it may be Ihown that, Q. E. D. io6 BOOK III. PROP. XXIII. THEOR. PON t/ie fame Jlraight line, and upon the fame fide of it, two fmilar fegments of cir- cles cannot be conflruBed which do not coincide. For if it be poflible, let two fimilar fegments o and be conftrudled ; draw any right line draw « • cutting both the fegments, and ^^-HMM . Becaufe the fegments are fimilar. (B. 3. def 10.), but ^M [Z ^^ (B. I. pr. 16.) which is abfurd : therefore no point in either of the fegments falls without the other, and therefore the fegments coincide. Q. E. D. BOOK III. PROP. XXIV. THEOR. 107 IMILAR fegments and , of cir- cles upon equal Jlraight lines ( '^^^ and — — ) are each equal to the other. For, if that — be fo applied to - may fall on ^— ^— may be on the extremities the extremities of and at the fame fide as becaufe muft wholly coincide with and the fimilar fegments being then upon the fame flraight line and at the fame fide of it, muft alfo coincide (B. 3. pr. 23.), and are therefore equal. Q. E. D. io8 BOOK III. PROP. XXV. PROP. SEGMENT of a circle being given, to defcribe the circle of 'which it is the feginent. From any point in the fegment draw ^^— ^ and — ^^^^ bifeft them, and from the points of biledlion draw and where they meet is the centre of the circle. Becaufe — ..__ terminated in the circle is bifedied perpendicularly by ^■"■■"^ , it palTes through the centre (B. 3. pr. i.), likewife ^a^^M^ pafles through the centre, therefore the centre is in the interfedlion of thefe perpendiculars. Q.E. D. BOOK III. PROP. XXVI. THEOR. 109 N egua/ circles the arcs O ""' o on which Jiand equal angles, whether at the centre or circum ference, are equal. Firfl, let ^^ at the centre. Then fince = mmm ^ O- /\ and ^♦;;.„ ■•\ have ■ ■■■■■■ and But ▲ =▲ (B. i.pr.4.). (B.3.pr. 20.); • • O '"' o are fimilar (B. 3. def. 10.) ; they are alfo equal (B. 3. pr. 24.) 110 BOOK III. PROP. XXVI. THEOR. If therefore the equal fegments be taken from the equal circles, the remaining fegments will be equal ; hence (ax. 3.); and .*, But if the given equal angles be at the circumference, it is evident that the angles at the centre, being double of thofe at the circumference, are alfo equal, and there- fore the arcs on which they fland are equal. Q. E. D. BOOK III. PROP. XXVn. THEOR. 1 1 1 N equal circles. O-O the angles and ^^ which Jiand upon equal arches are equal, whether they be at the centres or at the circumferences. For if it be poffible, let one of them ▲ be greater than the other and make ▲ \ = 4 /. V_^-" = **»n„..« (B. 3. pr. 26.) but V«^ = **♦.....•♦ (hyp.) .'. ^-i_ ^ := > fc^ _ ^ ^ ^ a part equal to the whole, which is abfurd ; .*, neither angle is greater than the other, and ,*, they are equal. Q. E. D. ••••.•■••• 112 BOOK III. PROP. XXVIII. THEOR. N equa/ circles o-o egual chords arches. cut off equal From the centres of the equal circles. draw and and becaufe c=o alfo (IW-) (B. 3. pr. 26.) and .0 = (ax. 3.) Q. E. D. BOOK III. PROP. XXIX. THEOR. 113 N equal circles nd ••-- ivhich fub~ \ ^ ^^ / the chords ^— -^^ and tend equal arcs are equal. If the equal arcs be femicircles the propofition is evident. But if not, let — ^^i^ . — — i^ , and be drawn to the centres ; becaufe and but ^— — ^ and (hyp.) (B.3.pr.27.); •■»...... and «•-' (B. I. pr. 4.); but thefe are the chords fubtending the equal arcs. Q. E. D. 114 BOOK III. PROP. XXX. PROB. O l>ife^ a given n- arc draw Draw make — ^ _L — ^^-■" , and it bifedls the arc. ■*•«•■•■ Draw •"••■"»■ and and — --— (confl:.), is common, (conft.) . (B. i.pr.4.) = y"*'\ (B. 3- pr. 28.). and therefore the given arc is bifeded. Q. E. D. BOOK III. PROP. XXXI. THEOR. 115 N a circle the angle in afemicircle is a right angle, the angle in a fegment greater than a femicircle is acute, and the angle in a feg- ment lefs than a femicircle is obtufe. FIGURE I. FIGURE I. The angle ^ in a femicircle is a right angle. V Draw and and V = ^ (B. i.pr. 5.) + A= V ^ the half of two right angles sz a right angle. (B. i. pr. 32.) FIGURE II. The angle ^^ in a fegment greater than a femi- circle is acute. FIGURE II. Draw the diameter, and .- ^ a right angle ^^ is acute. ii6 BOOK III. PROP. XXXI. THEOR. FIGURE III. FIGURE III. The angle ^^^^ in a fegment lefs than femi- circle is obtufe. Take in the oppofite circumference any point, to which draw «mmm* and . ^ Becaufe W^ -|- (B. 3. pr. 22.) ^Oh but (part 2.), is obtufe. Q. E. D. BOOK III. PROP. XXXIL THEOR. 117 F a rig/it line ^■■■■ii"— ■ be a tangent to a circle, and frotn the point of con- tact a right line " be drawn cutting the circle, the angle jg^ made by this line with the tangent is equal to the angle ^^ in the alter- ate fegment of the circle. If the chord fhould pafs through the centre, it is evi- dent the angles are equal, for each of them is a right angle. (B. 3. prs. 16, 31.) But if not, dra'V from the point of contadl, it muft pafs through the centre of the circle, (B. 3. pr. 19.) .-. ^ = ^ (B.3.pr.3i.) W + f =• CA = f (B- I- pr. 32.) /. ^ = ^ (ax.). Again CJ = iV\ = _ + ^ (B. 3. pr. 22.) ^ /. C. y = ^m » (ax.), which is the angle in the alternate fegment. Q. E. D. ii8 BOOK III. PROP. XXXIII. PROP. N a given ftraight line ^^^— to dejcribe a Jegment of a circle that Jliall contain an angle equal to a given angle ^,ty, If the given angle be a right angle, bifedt the given line, and defcribe a femicircle on it, this will evidently contain a right angle. (B. 3. pr. 31.) If the given angle be acute or ob- tufe, make with the given line, at its extremity. , draw make with f defcribe and or — ' ■ ■ ■ as radius, for they are equal. is a tangent to o (B. 3. pr. 16.) divides the circle into two fegments capable of containing angles equal to / W and j/^ which were made refpedlively equal and (B. 3.pr. 32.) Q. E. D. BOOK III. PROP. XXXIV. PROP. 119 O cut off from a given cir- cle o a fegment which fiall contain an angle equal to a given angle I>raw (B. 3. pr. 17.), a tangent to the circle at any point ; at the point of contad: make and > the given angle ; contains an angle ^ the given angle. Becaufe and « angle in > • IS a tangent, cuts it, the (B. 3. pr. 32.), but (conft.) Q. E. D. 120 BOOK III. PROP. XXXV. THEOR. FIGURE I. FIGURE II. F two chords \ •• ••"" i .^ ^ circle interfeSl each other, the reBangle contained by the fegments of the one is equal to the re 51 angle contained by the fegments of the other. FIGURE I. If the given right lines pafs through the centre, they are bifedled in the point of interfed:ion, hence the recftangles under their fegments are the fquares of their halves, and are therefore equal. FIGURE II. Let — "— - pafs through the 'centre, and .«■>■.■- not; draw — — — — and . Then X or » (B. 2. pr. 6.), X = ' X - = ■■ (B. 2. pr. 5.). X FIGURE III. FIGURE III. Let neither of the given lines pafs through the centre, draw through their interfedlion a diameter ........ 9 and X = X >■■■•■ (Part. 2.), alfo - X = X ........ (Part. 2.) ; Q. E. D. BOOK III. PROP. XXXFI. THEOR. 121 F from a point without a FIGURE I. circle twojlraight lines be drawn to it, one of which — ■'^"» is a tangent to the circle, and the other — — -- cuts it ; the re^angle under the whole cutting line — ••«■• and the external fegment ^-^ is equal to the fquare of the tangent -^— , FIGURE I. Let — i— •• pafs through the centre; draw from the centre to the point of contadl ; - (B. i.pr. 47), minus or mmus (B.2.pr. 6). FIGURE II. If ■"•'■ do not pafs through the centre, draw FIGURE n. and Tl len "X minus (B. 2. pr. 6), that is. mmus .2 (B. 3.pr. 18). Q. E. D. 122 BOOK in. PROP. XXXVII. THEOR. but F from a point outfide of a circle tivoftraight lines be draivn, the one -■^-■» cutting the circle, the other — — ^ meeting it, and if the reSiangle contained by the whole cutting line —"« and its ex- ternal fegment ■-..—.. be equal to thefquare of the line meeting the circle, the latter .m.^m^m,—> is a tangent to the circle. Draw from the given point ^— , a tangent to the circle, and draw from the centre .«»■», ••».•«.••, and — --- — -^ -■^ = X (B.3.pr.36.) 2 = X (i^yp-). and Then in ', and and and -,^^^ ...a»— and is common. but and .*. ^ = ^ (B. i.pr. 8.); ^ ^^^ a right angle (B. 3. pr. 18.), ^r := ^_J| a right angle, ■^ is a tangent to the circle (B. 3. pr. 16.' Q. E. D. BOOK IV. DEFINITIONS. I. RECTILINEAR figure is faid to be infcribed in another, when all the angular points of the infcribed figure are on the fides of the figure in which it is faid to be infcribed. II. A FIGURE is faid to be defcribed about another figure, when all the fides of the circumfcribed figure pafs through the angular points of the other figure. III. A RECTILINEAR figure is faid to be infcribed in a circle, when the vertex of each angle of the figure is in the circumference of the circle. IV. A RECTILINEAR figure is faid to be cir- cumfcribed about a circle, when each of its fides is a tangent to the circle. 124 BOOK IF. DEFINITIONS. V. A CIRCLE is faid to be tnfcribed in a redlilinear figure, when each fide of the figure is a tangent to the circle. VI. A CIRCLE is faid to be circum- fcribed about a redtihnear figure, when the circumference pafles through the vertex of each angle of the figure. y is circumfcribed. VII. A STRAIGHT line is faid to be tnfcribed in a circle, when its extremities are in the \ circumference. The Fourth Book of the Elements is devoted to the folution of J problems t chiefly relating to the infcription and circumfcrip- tion of regular polygons and circles. A regular polygon is one whofe angles and fides are equal. BOOK IF. PROP. I. PROP. 125 N a given circle O to place ajlraight line, equal to agivenjlraight line ( ), not greater than the diameter of the circle. Draw , the diameter of and if ..-....^— . ^:z ', then the problem is folved. But if be not equal to (hyp-) ; make (B. I. pr. 3.) with as radius. defcribe I ), cutting f |, and draw ^ which is the line required. For (B. I. def. 15. confl.) Q. E. D. 126 BOOK IF. PROP. II. PROP. N a given circle O to m- Jcribe a triangle equiangular to a given triangle. To any point of the given circle draw , a tangent (B. 3. pr. 17.) ; and at the point of contadt make ^^^ — - ^^ (B. i. pr. 23.) and in like manner draw Ik and Becaufe and J^ = ^ (conft.) Jg^ = ^^ (B. 3. pr. 32.) .*. ^^ = ^U ; alfo V^ =: ^r ^°^ ^^ i-3xtit reafon. ,\^ = ^ (B. i.pr. 32.), and therefore the triangle infcribed in the circle is equi- angular to the given one. Q^E. D. BOOK IF. PROP. III. PROB. 127 BOUT a given circle O to circumfcribe a triangle equi- angular to a given triangle. Produce any fide , of the given triangle both ways ; from the centre of the given circle draw any radius. Make ^ft = ^ and (B. I. pr. 23.) r=% At the extremities of the three radii, draw and .-.-...--, tangents to the given circle. (B. 3. pr. 17.) Zi The four angles of >^Wi ^B , taken together, are equal to four right angles. (B. i. pr. 32.) 128 BOOK IF. PROP. III. PROB. but ^B ^"d ^^^ ^^^ I'ight angles (confl.) two right angles but ^H^ ^ La^^^M^ (B' ^' P''- ^3-) and = ^^ (conft.) and ,*, In the fame manner it can be demonftrated that <^=^, 4 = 4 (B. i.pr. 32.) and therefore the triangle circumfcribed about the given circle is equiangular to the given triangle. Q, E. D. i BOOK IF. PROP. IV. PROB. 129 fcribe a circle. Bifeft ^ and ^V. (B. i.pr. 9.) by and "— ^^ from the point where thefe lines meet draw -•■■— , and »•••■ refpedlively per- pendicular to — BMI^HiB , In and / and > A 4 and common, ,*, «••••••.•. ^^ .■■■...•.». (B. i. pr. 4and 26.) In like manner, it may be fhown alfo that hence with any one of thefe lines as radius, defcribe and it will pafs through the extremities of the o other two ; and the fides of the given triangle, being per- pendicular to the three radii at their extremities, touch the circle (B. 3. pr. 16.), which is therefore infcribed in the given circle. Q. E. D. 130 BOOK IV. PROP. V. PROB. O defer ibe a circle about a given triangle. ■" and --- (B. I. pr. 10.) From the points of bifedlion draw — ■•■■•■■••• J_ -^■~— ^ and ' and refpec- tively (B. i. pr. 11.), and from their point of concourfe draw i^--^^^, «••■-—— and and defcribe a circle with any one of them, and it will be the circle required. In (conft.). - common, ^ (conft.), (B. I. pr. 4.). ■■■^•■■aiaKa In like manner it may be fhown that , , ■■■■««■■■■ ^^ ^■^■^^^^^^" m^^^ "^^^^ \ and therefore a circle defcribed from the concourfe of thefe three lines with any one of them as a radius will circumfcribe the given triangle. Q. E. D. BOOK IF. PROP. FI. PROB. 131 O N a given circle f j /<? infcribe a fquare. Draw the two diameters of the circle _L to each other, and draw o is a fquare. f For, iince ^^^^ and ^^^ are, each of them, in a femicirclc, they are right angles (B. 3. pr. 31), /. — ^ 11 (B. i.pr. 28): and in like manner And becaufe mg ^ |^^ (confl.), and •••■•■■SM« """ >■■>■■■■■■■ ""• •••»»•■•■••= f B. I . def. I c). .*. — = —> — (B. I. pr. 4); and fmce the adjacent fides and angles of the parallelo- gram ^ X are equal, they are all equal (B. i. pr. 34) ; o and /, -^ ^ , infcribed in the given circle, is a fquare. Q. E. D. 132 BOOK IF. PROP. VIL PROP. BOUT a given circle I i ^^ circumfcribe a fquare. Draw two diameters of the given circle perpendicular to each other, and through their extremities draw 1 9 9 tangents to the circle ; and — and D is a fquare. — / I a right angle, (B. 3. pr. 18.) alfo - II (conft.), 5 in the fame manner it can be demonftrated that that — ^^ and - • •■*••« ■■ !■ and alfo ,», I I is a parallelogram, and becaufe they are all right angles (B. i. pr. 34) it is alfo evident that — --^ , —— ^.^ , and -i^— ^ are equal. D is a fquare. Q. E. D. BOOK IV. PROP. Fill. PROB. 133 O infcribe a circle in a given fquare. Make and draw and — — II .. (B. I. pr. 31.) and fince is a parallelogram ; is equilateral (B. i. pr. 34.) In like manner, it can be fhown that are equilateral parallelograms ; and therefore if a circle be defcribed from the concourfe oi thefe lines with any one of them as radius, it will be infcribed in the given fquare. (B. 3. pr. 16.) CLE. D. 134 BOOK IF. PROP. IX. PROB. O defcribe a circle about a given fquare 3 Draw the diagonals ^— — .-. and "— ■ interfedting each other ; then, becaufe "^-^Ik have their fides equal, and the bafe ■ ■»«*»■• comnion to both, or r ^ (B. i.pr. 8), is bifedled : in like manner it can be (hown that is bifedted ; hence ^k rr ^^ their halves. •. = ; (B. I. pr. 6.) and in like manner it can be proved that If from the confluence of thefe lines with any one of them as radius, a circle be defcribed, it will circumfcrihe the given fquare. Q. E. D. BOOK IV. PROP. X. PROP. '35 O conftruSl an ifofceles triangle, in which each of the angles at the bafejliail he double of the vertical angle. Take any fliaiwht line ^— and divide it fo that X = (B. 2. pr. I I.) With —I"" as radius, defcribe o in it from the extremity of the radius, (B. 4. pr. i) ; draw Then \ and place \ is the required triangle. For, draw and defcribe O Since about ^ I (B. 4. pr. ^.) X • ■■■■■ X "■ •— is a tangent to ( ) (B. 3. pr. 37.) .% m = ^ (B. 3. pr. 32), 1 36 BOOK IV. PROP. X. PROB. add ^F to each, ••• A + < = ^ + ^; but ▼ + A or # =z A (B. I. pr. 5) : fince -—"m-m ^ ■"»» (B. I. pr. 5.) confequently jH[^ ^ ^Xi ^ ^f ^ JH^ (B. I. pr. 32.) .'. — = (B. i.pr. 6.) .•. — ^— =z — — ^— ^ .^_. (conft.) .-. -^ = W (B. I.pr. 5.) .-. A=^ = A = ^ + =: twice y^t 5 and confequently each angle at the bafe is double of the vertical angle. Q. E. D. BOOK IV. PROP. XL PROB. 137 N a given circ/e o to infcribe an equilateral and equi- angular pentagon. Conftrudl an ifofceles triangle, in which each of the angles at the bafe fhall be double of the angle at the vertex, and infcribe in the given ▲ circle a triangle ^^ equiangular to it ; (B. 4. pr. 2.) ^ and ^\ (B. i.pr.9.) Bifedl draw and Becaufe each of the angles A.^.A ^ and \\ are equal, the arcs upon which they ftand are equal, (B. 3. pr. 26.) and .*. and ....■■». which fubtend thefe arcs are equal (B.3.pr. 29.) and ,*, the pentagon is equilateral, it is alfo equiangular, as each of its angles ftand upon equal arcs. (B. 3. pr. 27). Q^E. D. 138 BOOK IV. PROP. XII. PROP. O defcribe an equilateral and equiangular penta- gon about a given circle O- Draw five tangents through the vertices of the angles of any regular pentagon infcribed in the given circle o (B. 3. pr. 17). Thefe five tangents will form the required pentagon. Draw ' 1:™ In and ■ ^■•■■■■■B (B. i.pr.47), and ■ common ; .-.7 = <- twice and ▼ = .4. (B. i.pr. 8.) , and ^^1 =r twice ^ In the fame manner it can be demonilrated that ^^/ =: twice ^^ , and ^r ^ t^vice ^; but ^ ^ " B. 3.pr. 27), BOOK IV. PROP. XII. PROB. 139 their halves = j^ , alfo £ I ^ I \ 9 and -■■■-■■■» common ; and «-i-iaMaiii. ^ ...HMMiB, ,•, •■^-■» .— ^K ^ twice — ^— 5 In the fame manner it can be demonftrated that 1^^^---— ^ twice ■-■^-•, but — — = — — • In the fame manner it can be demonftrated that the other fides are equal, and therefore the pentagon is equi- lateral, it is alfo equiangular, for ^^ ^ twice 1^^ and \^^ = twice j^^ , and therefore • mKkl — uflB 9 1" the fame manner it can be demon ftrated that the other angles of the defcribed pentagon are equal. QE. D 140 BOOK IF. PROP. XIII. PROP. Draw Becaufe and O infcribe a circle in a given equiangular and equilateral pentagon. ^^^ «■/ ^^ ^ given equiangular and equilateral pentagon ; it is re- quired to infcribe a circle in it. Make ^=^,andi|^=^ (B. i.pr. 9.) = - ,r=A, common to the two triangles &c. / and >A ...,.lk ; .. and ^r ^ J|^ (B. i. pr. 4.) And becaufe ^^ ^ ,*, r= twice 4 twice is bifedled by In like manner it may be demonftrated that ^ IS «••• "j and that the remaining angle of bifedled by the polygon is bifedled in a fimilar manner BOOK IV. PROP. XIII. PROB. 141 Draw ^i— -i^ , -....-.. , 6cc. perpendicular to the fides of the pentagon. Then in the two triangles ^^ and A we have ^^ z= ^^^,(conft.), ^^^^i^ common, and ^V :^ JIh = a right angle ; , (B. I. pr. 26.) In the fame way it may be fhown that the five perpen- diculars on the fides of the pentagon are equal to one another. o Defcribe X^ ^ with any one of the perpendicu- lars as radius, and it will be the infcribed circle required. For if it does not touch the fides of the pentagon, but cut them, then a line drawn from the extremity at right angles to the diameter of a circle will fall within the circle, which has been fhown to be abfurd. (B. 3. pr. 16.) f^E. D. 14* BOOK IV, PROP. Xn\ PROB. pO dcfirihc j r.-TiV chcn: s grom egh:.s: 'sJ ^nd csiii oik and ^^ Bilect ^JHk and bT ••••••»»»«» and ..-•...... , and ^om the point of fedion, draw ._^B , >••»»• , and ^^^^ . (B. i.pr.6): I" like manner it mar be proved that ■ ^ ^iB^^M ^ ^^— — , and therefore -••••••• ^ — ^— ^ ••m>»...w. Therefore if a circle be defcribed from the point where thefe five lines meet, with any one of tfaem as a radius, it will circumicribe the given pcntagoo. Q E- D. BOOK W. PROP. XV. PROB. O infcribe an equilateral and equian- gular hexagon in a gircen circle H3 O- From any point in the circumference of the given circle defcribe ^ J palling o through its centre, and draw the diameters and draw ...._»■_ J .-..-..-^ .........J &c. and the required hexagon is inicribed in the given circle. Since of the circles. palles through the centres and are equilateral 4 = ^ triangles, hence ^^ ^ ^^ ^ one-third of two right angles; i^B. i. pr. 32) but (B. i.pr. 13); = m ^ one-third of £Di (B. I. pr. 32% and the angles vertically oppoiite re :::ei"e are all equal to one another (B. i. pr. i ;\ and iland on equal arches (B. 3. pr. 26), which are fubtended by equal chords (B. 3. pr. 29) ; and fince each of the angles of the hexagon is double of the angle of an equilateral triangle, it is alio equiangular. O ^ F) '44 BOOK IV. PROP. XVI. PROP. O infcribe an equilateral and equiangular quindecagon in a given circle. and be the fides of an equilateral pentagon infcribed in the given circle, and ««»-— the fide of an inscribed equi- lateral triangle. The arc fubtended by . and __ _6_ I 4 of the whole circumference. The arc fubtended by 1 _5_ 1 i Their difference =: tV ,'. the arc fubtended by the whole circumference. of the whole circumference. zz. tV difference of Hence if firaight lines equal to ..-«.—■« be placed in the circle (B. 4. pr. i), an equilateral and equiangular quin- decagon will be thus infcribed in the circle. Q. E. D. BOOK V. DEFINITIONS. I. LESS magnitude is faid to be an aliquot part or fubmultiple of a greater magnitude, when the lefs meafures the greater ; that is, when the '^ lefs is contained a certain number of times ex- adlly in the greater. II. A GREATER magnitude is faid to be a multiple of a lefs, when the greater is meafured by the lefs ; that is, when the greater contains the lefs a certain number of times exadlly. III. Ratio is the relation which one quantity bears to another of the fame kind, with refpedl to magnitude. IV. Magnitudes are faid to have a ratio to one another, when they are of the fame kind ; and the one which is not the greater can be multiplied fo as to exceed the other. TAe of her definitions will be given throughout the book where their aid is Jirjl required. u 146 AXIOMS. QUIMULTIPLES or equifubmultiples of the fame, or of equal magnitudes, are equal. If A = B, then twice A ^ twice B, that is, 2 A = 2 B; 3Az=3B; 4 A = 4B; &c. &c. and i of A = i of B ; i of A = i of B ; &c. &c. II. A MULTIPLE of a greater magnitude is greater than the fame multiple of a lefs. Let A C B, then 2 AC 2 B; 3 ACZ3B; 4 AIZ4B; &c. &c. III. That magnitude, of which a multiple is greater than the fame multiple of another, is greater than the other. Let 2 A C 2 B, then ACB; or, let 3 A C 3 B, then ACB; or, let w A CZ m B, then ACB. \ BOOK V. PROP. I. THEOR. H7 F any number of magnitudes be equimultiples of as many others, each of each : what multiple soever any one of the firjl is of its part, the fame multiple jhall of the fir ft magnitudes taken together be of all the others taken together. LetQQQQQ be the fame multiple of Q, that Pip^^^ isof ^. that OOOQO ^s of Q. Then is evident that • QQQQQ OOOOQ fQ is the fame multiple of < Q which that QQQQQ is of Q ; becaufe there are as many magnitudes QQQQQ 1 m <! .QOOQOJ Q Q as there are in QQQQQ := Q. The fame demonftration holds in any number of mag- nitudes, which has here been applied to three. ,*, If any number of magnitudes, &c. 148 BOOK r. PROP. 11. THEOR. |F the fir ft magnitude be the fame multiple of the fecondthat the third is of the fourth, and the fifth the fame multiple of the fecond that the fixth is oj the fourth, then fiall the firfi, together with the fifth, be the fame multiple of the fecond that the third, together •with the fixth, is of the fourth. Let ^01 9, the firil, be the fame muhiple of ^, the fecond, that OO 0> ^'^^ third, is of <^, the fourth ; and let 9 0, the fifth, be the fame multiple of * , the fecond, that OOOOj ^^^ fixth, is of <2>, the fourth. Then it is evident, that \ .^ ,^. ,^. .^ ' , the firfl and fifth together, is the fame multiple of , the fecond, that \ !r!r^^ k the third and fixth together, is of looooj the fame multiple of <2>) the fourth ; becaufe there are as f #•• 1 many magnitudes i" ] ^^^^ ^ 3P ^s there are • f 000 \ _ ^ ,*, If the firfl: magnitude, &c. BOOK V. PROP. III. THEOR. 149 F thefirjl of four magnitudes be the fame multiple of the fecond that the third is of the fourth, and if any equimultiples whatever of the firjl and third be taken, thofe Jliall be equimultiples ; one of the fecond, and the other of the fourth. The First. The Second. Let \ " |- be the iame multiple of I ! The Third. The Fourth. which j T T [ is of A ; take <;^ S S S S > the fame multiple of < which \ J 3 ? A [♦♦♦♦ > is of \ ♦♦ that ^ Then it is evident, Tlie Second. is the fame multiple of | 150 BOOK V. PROP. III. THEOR. which i ♦♦♦♦ ♦ ♦♦♦ ♦ ♦♦♦ The Fourth. - is of ^ ; J becaufe < > contains > contains ♦♦♦♦' ♦♦♦♦ ♦♦♦♦ as many times as ;- contains ■! T^ T^ !> contains ^ ♦♦ The fame realbning is applicable in all cafes. .*. If the firft four, &c. BOOK V. DEFINITION V. '51 DEFINITION V. Four magnitudes, ^^ 01 ^ ^ > ^j are faid to be propor- tionals when every equimultiple of the firft and third be taken, and every equimultiple of the fecond and fourth, as. of the firfl &c. of the fecond of the third ^ ^ ♦ ♦♦♦ ♦ ♦♦♦♦ ♦♦♦♦♦♦ &c. of the fourtli &c. &c. Then taking every pair of equimultiples of the firft and third, and every pair of equimultiples of the fecond and fourth, ' — =or^«« = or 31 or ^ = or 33 = or — 1 r \^<mm cz, = L ^♦4 C. = °r ♦ ♦ C. = or then will - ♦ ♦ C, = or tt C. = or ^♦^ C. = or 152 BOOK F. DEFINITION V. That is, if twice the firfl be greater, equal, or lefs than twice the fecond, twice the third will be greater, equal, or lefs than twice the fourth ; or, if twice the firfl be greater, equal, or lefs than three times the fecond, twice the third will be greater, equal, or lefs than three times the fourth, and so on, as above expreffed. If or or or or or then will &c. '♦♦♦ ♦♦♦ ♦♦♦ ♦ ♦♦ ♦ ♦♦ &c. c = 6cc. or Z] or 31 or ^ or Z] or ^ &c. In other terms, if three times the firft be greater, equal, or lefs than twice the fecond, three times the third will be greater, equal, or lefs than twice the fourth ; or, if three times the firft be greater, equal, or lefs than three times the fecond, then will three times the third be greater, equal, or lefs than three times the fourth ; or if three times the firft be greater, equal, or lefs than four times the fecond, then will three times the third be greater, equal, or lefs than four times the fourth, and so on. Again, BOOK V. DEFINITION V. ^Sl If < Sec. then will ♦ ♦♦♦ ♦ ♦♦♦ ^ or ^ = or ^ = or Zl =: or ID = or ID = or Zl = or Zl = or Z] = or Zl = or Zl &c. &c. &;c. And so on, with any other equimultiples of the four magnitudes, taken in the fame manner. Euclid exprefles this definition as follows : — The firft of four magnitudes is faid to have the fame ratio to the fecond, which the third has to the fourth, when any equimultiples whatfoever of the firft and third being taken, and any equimultiples whatfoever of the fecond and fourth ; if the multiple of the firft be lefs than that of the fecond, the multiple of the third is alfo lefs than that of the fourth ; or, it the multiple of the firft be equal to that of the fecond, the multiple of the third is alfo equal to that of the fourth ; or, ii the multiple of the firft be greater than that of the fecond, the multiple of the third is alfo greater than that of the fourth. In future we ftiall exprefs this definition generally, thus : If M # C = or Zl ''^ , when M ^ C = or 313 ;;; ^ 154 BOOK V. DEFINITION F. Then we infer that , the firft, lias the lame ratio to ^ , the fecond, which ^ , the third, has to ^ the fourth : exprelTed in the fucceeding demonftrations thus : # :ii :: 4 : V; or thus, : It = ^ • V 7 or thus, — ^ =■ : and is read, " as is to , so is ^ to ^. And if # : " : : ^ : ip we fhall infer if M C5 ^= or ^ w , , , then will M ^ C = or 13 w ^. That is, if the firfl; be to the fecond, as the third is to the fourth ; then if M times the firft be greater than, equal to, or lefs than m times the fecond, then (hall M times the third be greater than, equal to, or lefs than m times the fourth, in which M and m are not to be confidered parti- cular multiples, but every pair of multiples whatever; nor are fuch marks as 0, ^, , &c. to be confidered any more than reprefentatives of geometrical magnitudes. The ftudent fhould thoroughly underftand this definition before proceeding further. BOOK V. PROP. IF. THEOR. ^SS F the firjl of four magnitudes have the fame ratio to the fecond, which the third has to the fourth, then any equimultiples whatever of the frji and third shall have the fame ratio to any equimultiples of the fecond and fourth ; viz., the equimultiple of the firji Jliall have the fame ratio to that of the fecond, which the equi- multiple of the third has to that of the fourth. m Let : ■ ::^ :^, then3 :2|::3^:2^, every equimultiple of 3 and 3 ^ are equimultiples of ^ and ^ , and every equimultiple of 2 ^ and 2 ^ , are equimultiples of | and ^ (B. 5, pr. 3.) That is, M times 3 '^ and M times 3 ^ are equimulti- ples of and ^ , and ;;z times 2 | and w 2 1^ are equi- multiples of 2 H and 2 ^ ; but • H • • ^ • V (hyp); .*. if M 3 C =, or ;^ »? 2 |||, then M 3 ^ C r=, or Z] « 2 ip (def. 5.) and therefore 3 ^:2||::3^:2^ (def. 5.) The fame reafoning holds good if any other equimul- tiple of the firft and third be taken, any other equimultiple of the fecond and fourth. ,*. If the firfl: four magnitudes, &c. 156 BOOK V. PROP. V. THEOR. F one magnitude be the fame multiple of another, which a magnitude taken from thefirfl is of a mag- nitude taken from the other, the remainder Jhall be the fame multiple of the remainder, that the whole is of the whole. Q LetQQ O = M'^ and = M-,, o C^<^ minus = M' minus M' «> D .-. <> =M'(Jminus.), and /. ^ = M' A. /, If one magnitude, Sec. BOOK V. PROP. VI. THEOR. 157 F two magnitudes be equimultiples of two others, and if equimultiples of thefe be taken from the fir Ji two, the remainders are either equal to thefe others, or equimultiples of them. Q Let :yQ = M' ■ ; and QQ = M' a ; o Q then 00 minus ni m ::^ o M' ■ minus /w' « = (M' minus m') b, and 00 minus m' k := M' a minus /«' 4 := (M' minus m') k . Hence, (M' minus ;«') m and (M' minus tn') k are equi- multiples of K and k , and equal to * and a 9 when M' minus m' ':^i i. .'. If two magnitudes be equimultiples, &c. 158 BOOK F. PROP. A. THEOR. F the firjl of t/ie four magnitudes has the fame ratio to the fecond which the third has to the fourth, then if the firjl be greater than the fecond, the third is alfo greater than the fourth ; and f equal, equal ; if lefs, lefs. Let ^ : H : r ip : ^ ; therefore, by the fifth defini- tion, if %% d ■■, then will ^^ C #4 ; but if # CZ ■, then ## [Z ■■ and ^fp [= and .*. ^ C ;► . Similarly, if ^ ^, or ^ J, then will ^ z^, or ^ . .'. If the firfl of four, &c. DEFINITION XIV. Geometricians make ufe of the technical term " Inver- tendo," by inverfion, when there are four proportionals, and it is inferred, that the fecond is to the firfl as the fourth to the third. Let A : B : : C : O, then, by " invertendo" it is inferred B : A :: I) : C. BOOK V. PROP. B. THEOR. '50 F Jour magnitudes are proportionals , they are pro- portionals alfo when taken inverfely. Let ^ : O : : ■ : ^ , then, inverfely, O : ^ 1 1 : ■ . If M ^ n « O? then M ■ I] w ^ by the fifth defimtion. Let M ^ ID /w Q, that is, w Q CZ M ^ , .*. M B lU w , or, /« CZ M ■ ; .*. if w O CZ M ^, then will w C M B In the fame manner it may be (liown, that if ;« Q := or Z] M ^ , then will m :=, or 13 M B ; and therefore, by the fifth definition, we infer that O : ^ : '^ : H . .', If four magnitudes, &c. i6o BOOK V. PROP. C. THEOR. F the fiyji be the fame multiple of the fecond, or the fame part of it, that the third is of the fourth ; the frjl is to the fecond, as the third is to the fourth. Let ^ ^ t the firfl:,be the fame multiple of ^, the fecond, that 7 J, the third, is of ■, the fourth. ♦ ♦.4 ♦ ♦ ,m 0, M ? ?,>« A Then _ _ takeMj J ■ ■ becaufe^S is the fame multiple of ^ that J J is of 4 (according to the hypothcfis) ; and ^M^ is taken the fame multiple ofSS that M T T is of ? T , ,*, (according to the third propolition), M ^ _ is the fame multiple of ^ that M T T is of 4. BOOK F. PROP. C. THEOR. i6i Therefore, if M ^ ^ be of ^ a greater multiple than ;// ^ is, then M J i is a greater multiple of ^ than w A is ; that is, if M S S ^^ greater than m 0, then M J J will be greater than m ^ ; in the fame manner it can be fliewn, if M ^ ^ be equal m ^^ then M J J will be equal m A. And, generally, if M ^ ^ C =z or ^ //; then M will be C ^ or ^ ;« ^* ,', by the fifth definition, ■ ■_..♦♦.▲ ■ ■•••♦♦•■• ■ ■ Next, let be the fame part of J S that itk is of T T- In this cafe alfo : J J :: (ffc : TT. For, becaufe ■■"*■"-""' WW is the fame part of ^ ^ that ■ is of ^ ^ , 1 62 BOOK F. PROP. C. THEOR. therefore S S is the fame muhiple of that ^ J is of ^ . Therefore, by the preceding cafe, ■ ■ . A .. . ^ . and • ^ • ■■ •• A • ^^ by propofition B. /. If the firfl be the fame multiple, &c. BOOK V. PROP. D. THEOR. 163 \^ the fir Jl be to the fecond as the third to the fourth, and if the firfi be a multiple, or a part of the fecond; the third is the fame multiple, or the fame part of the fourth. and firft, let be a multiple H; J J fhall be the fame multiple of ■. First. Second. Tliird. Fourth. QQ 00 Take ^^ =r ^ QQ Whatever multiple take ^^ then, becaufe is of I the fame multiple of 1 • ....♦♦ and of the fecond and fourth, we have taken equimultiples, ^nd )f Y , therefore (B. c. pr. 4), 00 i64 BOOK F. PROP. D. THEOR. 'OCl"^^ . ^^, but (conft.), and y\y\ is the fame multiple of ■ that ^ is of U . Next, let B : ^ ^ • ■ V • T J » and alfo H a part of -^^ ; then ip fhall be the fame part of J J , Inverfely (B. 5.), ^ ' ■ *= ^ J ' V' but I is a part of ^^ ; that is, ^ ^ is a multiple of | ; , by the preceding cafe, X X is the fame multiple of ^ that is, ^ is the fame part of X X that H is of ,% If the firft be to the fecond, &c. BOOK V. PROP. VII. THEOR 165 QUAL magnitudes have the fame ratio to the fame tnagnitiide, and the fame has the fame ratio to equal magnitudes. Let ^ = ^ and any other magnitude ; then : = ♦ •* and : # = : : ^ , Becaufe ^ ^ ^, .-. M # = M ^ ; /, if M C := or [3 /;/ , then M ^ C = or ;^ ;« , and .*. % : c = ♦ : ■ (B. 5. def. 5). From the foregoing reafoning it is evident that, i£ m C > = Of ZD M , then wHC^orl^ M^ /.■:#=■:♦ (B. 5. def. 5). ,*. Equal magnitudes, &c. i66 BOOK F. DEFINITION VII. DEFINITION VII. WiiKN of the equimultiples of four magnitudes (taken as in tile fifth definition), the multiple of the firft is greater than tli:it of the fecond, but the multiple of the third is not greater than the multiple of the fourth ; then the firft is laid to have to the fecond a greater ratio than the third magnitude has to the fourth : and, on the contrary, the third is faid to have to the fourth a lefs ratio than the firft has to the fecond. If, among the equimultiples of four magnitudes, com- pared as in the fifth definition, we fhould find • #••# [=■■■■, but 44^44 = '"' ^ W IP V. or if we mould rnul .iny particular multiple M oi the firft and third, and a particular multiple m' of the fecond and fourth, fuch, that M times the firft is C w' times the fecond, but M' times the third is not CZ w times the fourth, i.e. ^ or ~~1 "; times the fourth ; then the firll is faid to have to the tiwnd a strcater ratio than the third has to the fourth; v>r the thial has to the fourth, under fuch circumftances, a lets ratio than the hrtl has to the feccaid : although feveral other equimultiples may tend to ibow that the four mag- nitudes arc piv>portionAls. This det\i\itivM\ will in tuiure be exprdSbd ti^is : — ItM fP C rr Q. but M ■ = - Z3 'T ♦ , then P : ~ IZ ■ : ♦ . In the aK>\^ cv' ;- ' ;\ -tSoia* M «ad af aie to be wnikkitHi jvftrtkn;. cs, at* fike dK iMilli|ilr' M BOOK F. DEFINITION VII. 167 and m introduced in the fifth definition, which are in that definition confidered to be every pair of multiples that can be taken. It muft alfo be here obferved, that ip , U, H , and the like fymbols are to be confidered merely the repre- fentatives of geometrical magnitudes. In a partial arithmetical way, this may be fet forth as follows : Let us take the four numbers, 8 , 7, j c , and FirJi. Second. Third. Fourth. 8 7 10 Q lO 14 2C» 24 21 30 ^7 32 28 40 36 40 35 50 45 48 42 60 54 56 49 70 63 64 56 80 72 72 63 90 8t 80 70 lOD ".'- 88 77 no vy 96 84 120 108 104 9' '3° 117 T12 98 140 126 &c. &c. &c &c. Among the above multiples we find 16 C 14 and 20 r~ that is, twice the firft is greater than twice the fecond, and twice the third is greater than twice the fourth ; and 16^21 and 20 "^ that is, twice the firil is lefs than three times the fecond, and twice the third is lefs than three times the fourth ; and among the fame multiples we can find -: C 56 and V - C that is, 9 times the firft is greater than 8 times the fecond, and 9 times the third is greater than 8 times the fourth. Many other equimul- 1 68 BOOK V. DEFINITION VII. tiples might be selected, which would tend to Ihow that the numbers %,y, \o, were proportionals, but they are not, for we can find a multiple of the firlt ^ a multiple of the fecond, but the fame multiple of the third that has been taken of the firft not C the fame multiple of the fourth which has been taken of the fecond ; for inftance, 9 times the hrll: is C i o times the fecond, but 9 times the third is not C ^° times the fourth, that is, -: C 70, but 90 not CZ or 8 times the firfl we find C 9 times the fecond, but 8 times the third is not greater than 9 times the fourth, that is, O-i-C 63, but Sc is not C When any fuch multiples as thefe can be found, the hrft (3~)is faid to have to the fecond (7) a greater ratio than the third (10) has to the fourth and on the contrary the third (10) is faid to have to the fourth a lefs ratio than the firfl (3) has to the fecond (7). BOOK r. PROP. Fill. THEOR. 109 F unequal magnitudes the greater has a greater ratio to the fame than the lefs has : and the fame magnitude has a greater ? atio to the lefs than it has to the greater. Let m and be two unequal magnitudes, and ^ any other. k We fliall firft prove that H which is the greater of the two unequal magnitudes, has a greater ratio to than , the lefs, has to ^ ^ that is, ■ : [Z , : # ; take M' l^/^' #, M' ■, and tn % ; fuch, that M' ▲ and M' H fhall be each C # ; alfo take ;;/ ^ the leaft multiple of ^ , which will make m M' = M' .*. M' is not CZ f" butM' IS ?n for. as m' A is the firft multiple which firft becomes C M'^, than (w minus I ) ^ or;;/ minus ^ isnotCM' JU, and ^ is not CI M' a, /. ;;;' minus + # "^"^ be Zl M' Jj + M' A ; A that is, ;;;' % mull be i;;^ M' ■ ; .-. M' IS tn ', but it has been fhown above that 170 BOOK F. PROP. Fill. THEOR. M' m is note »?' # , therefore, by the feventh definition, m has to a greater ratio than 1:0. Next we fhall prove that % has a greater ratio to ^ , the lefs, than it has to j^ , the greater ; o''# :■ [= • :■• A Take /;/ 0, M' ■■, m' #, and M' ||, the fame as in the firll: cafe, fuch, that M' A and M' jp will be each C , and m % the leail multiple of ^ , which firft becomes greater than M' H = M' || . .'. m ^ minus ^ is notC M' ^, and ^ is not CI M' A ; confequently m % minus # -}- # is ZH M' g -f M' a ; ▲ ,*, m' ^ is ID M' ■, and ,'. by the feventh definition, A ^ has to a| ^ greater ratio than ^ has to ■ . ,'. Of unequal magnitudes, &c. The contrivance employed in this propofition for finding among the multiples taken, as in the fifth definition, a mul- tiple of the firfl greater than the multiple of the fecond, but the fame multiple of the third which has been taken of the firft, not greater than the fame multiple of the fourth which has been taken of the fecond, may be illuftrated numerically as follows : — The number 9 has a greater ratio to 7 than has to 7 : that is, 9 : 7 C : 7 ; or, 8 + i : 7 C = 7- BOOKF. PROP. Fill. THEOR, 171 The multiple of i , which firft becomes greater than 7, is 8 times, therefore we may multiply the firft and third by 8, 9, 10, or any other greater number; in this cafe, let us multiply the firft and third by 8, and we have '^-^-f- 8 and : again, the firft multiple of ^ which becomes greater than 64 is 10 times; then, by multiplying the fecond and fourth by 10, we ftiall have 70 and 70 ; then, arranging thefe multiples, we have — 8 times lo times 8 times lo times the first. the second. the third. the fourtli. 6^+ 8 -0 -o Confequently 04 -j- 8, or 72, is greater than -o, but -t^ is not greater than 70, .•. by the feventh definition, 9 has a greater ratio to 7 than has to 7 . The above is merely illuftrative of the foregoing demon- ftration, for this property could be fhown of thefe or other numbers very readily in the following manner ; becaufe, if an antecedent contains its confequent a greater number of times than another antecedent contains its confequent, or when a fraction is formed of an antecedent for the nu- merator, and its confequent for the denominator be greater than another fraction which is formed of another antece- dent for the numerator and its confequent for the denomi- nator, the ratio of the firft antecedent to its confequent is greater than the ratio of the laft antecedent to its confe- quent. Thus, the number 9 has a greater ratio to 7, than 8 has to 7, for ^ is greater than -. Again, 17 : 19 is a greater ratio than 13 : 15, becaufe 17 17 X 15 _ 255 J 13 13 X 19 247 , evident that ^ is greater than |g, .-. J-^ is greater than 1/2 BOOK F. PROP. VIIT. THEOR. — , and, according to wliat has been above fliown, 17 has to 19 a greater ratio than 13 has to 15. So that the general terms upon which a greater, equal, or lefs ratio exifVs are as follows : — A C . . If g be greater than ^, A is faid to have to B a greater A C ratio than C has to D ; if — be equal to rr, then A has to B the fame ratio which C has to D ; and if -^ be lefs than ^, A is faid to have to B a lefs ratio than C has to D. The ftudent fhould underftand all up to this propofition perfectly before proceeding further, in order fully to com- prehend the following propofitions of this book. We there- fore ftrongly recommend the learner to commence again, and read up to this flowly, and carefully reafon at each ftep, as he proceeds, particularly guarding againlT; the mifchiev- ous fyftem of depending wholly on the memory. By fol- lowing thefe inftrudions, he will find that the parts which ufually prefent confiderable difficulties will prefent no diffi- culties whatever, in profecuting the ftudy of this important book. BOOK V. PROP. IX. THEOR. ^71, AGNITUDES which have the fame ratio to the fame magnitude are equal to one another ; and thofe to which the fame magnitude has t/ie fame rat to are equal to one another. Let ^ : ^ : : : p, then ^ = . For, if not, let ▲ CI ? then will 4 : «^ C # : (B. 5- pr. 8), which is abfurd according to the hypothefis. .*. ^ is not C . In the fame manner it may be fhown, that A is not ^ ▲, /. 4 =#. Again, let H : ^ : : '^' : ^ , then will ^ = . For (invert.) ^ : || : : f| : H, therefore, by the firft cafe, A ^ A . ,*. Magnitudes which have the fame ratio, &c. This may be {hown otherwife, as follows : — Let ^ : B ^ A : C, then B = C, for, as the fradlion — = the fradlion -, and the numerator of one equal to the numerator of the other, therefore the denominator of thefe fradlions are equal, that is K zz C. Again, if B : ,\ = C : A , B = C. For, as - = "^, B muft = (,. 174 BOOK V. PROP. X. THEOR. HAT magnitude which has a greater ratio than another has unto the fame magnitude, is the greater of the two : and that magnitude to which the fame has a greater ratio than it has unto another mag- nitude, is the lefs of the two. Let ^ : C # : ■> then ^ d # . For if not, let |p =: or ^ ; then, ^ : si = # : B (^- 5- P^- l) or ^ : H 13 ^ : ■ (B. 5. pr. 8) and (invert.), which is abfurd according to the hypothefis. ,*, ■ is not =: or ^ ^ , and .'. S muft he r~ ^. Again, let «: C V : fP, then, ID ^> For if not, mufl: be C or ^ 1^ , then flj: Z] p: ^ (B- 5. pr. 8) and (invert.); or fl: =: H* V (B. 5. pr. 7), which is abfurd (hyp.); /. is not CZ or = ^ , and .'. mufl be ^ ^ . .*. That magnitude which has, &c. BOOK V. PROP. XL THEOR. ^75 ATIOS t/iat are the fame to the fame ratio, are the fame to each other. Let ^ : ■ = : IP' and : P = ▲ : •, then will ^ : H = ▲ : •. For if M ^ C =, or 13 m H, then M IZ> ^. or 3] w ^, and if M CZ, ^, or ^ ;/; t' , then M A C =, or Z3 m •, (B. 5. def. 5) ; , if M ^ C, ^, or ^ w d 9 M A [Z, =, or Zl ^« •> and .*. (B. 5. def. 5) ^ I H ^ A : •. .*, Ratios that are the fame, &c. 176 BOOK r. PROP. XII. THEOR. F afijf number of tnagnitiides be proportionals, as one of the antecedents is to its confeqiient, fo f}:>all all the antecedents taken together be to all the confequents. then will | : # = ■ +0+ +« + ^:# + 0+ +' + •• For if M U IZ w 0, then M Q [Z w <>, and M \^tn M • CZ ^« t, alfoM ▲ IZ'« •• (B. 5. def. 5.) Therefore, if M JH C w 0, then will MJ+MQ + M -I-M. + Ma, or M (H + O + + • + ^) be greater tlian ;;/ ^ •\- tn ^ -\- m •\' ^'^ ▼ "h ^^^ •> or;^(# +0+ , + ^ + *)- In the fame way it may be fhown, if M times one of the antecedents be equal to or lefs than m times one of the con- fequents, M times all the antecedents taken together, will be equal to or lefs than ni times all the confequents taken together. Therefore, by the fifth definition, as one of the antecedents is to its confequent, fo are all the antecedents taken together to all the confequents taken together. .*, If any number of magnitudes, &c. BOOK V. PROP. XIII. THEOR. '77 F the firjl has to the fecond the fame ratio which the third has to the fourth, but the third to the fourth a greater ratio than the fifth has to the fixth ; the firfi fhall afo have to the fecond a greater ratio than the fifth to the fixth. Let fP : O = ■ : # , but ■ : A ci O : #, then fP : D IZ O : •• For, becaufe | : d O • 0> there are fome mul- tiples (M' and ;«') of | and ^^ and of ^ and ^, fuch that M' m but M' <^ not C m 0, by the feventh definition. Let thefe multiples be taken, and take the fame multiples of ■ and ([n. /. (B. 5. def. 5.) if M' ^ C, =, or Zl /«' Q ; then will M' | C =, or ^ m , but M' ■ C »^' ^ (conftrudlion) ; ■ .*. M' ^ C ni Q , but M' <3 is not C ni (conflrudlion) ; and therefore by the feventh definition, W :0 CIO ^^v * .*. If the firll; has to the fecond, &c. A A 178 BOOK V. PROP. XIV. THEOR. F the fir Jl has the fame ratio to the fecondivhich the third has to the fourth ; then, ifthefirjl be greater than the third, thefecondjhall be greater than the fourth; and if equal, equal; andiflefs, lefs. Let ^ : Q : : B : ^ , and firfl fuppofe IP [Z » , then will Q C ^ • For^rQCI : IJ (B.5.pr. 8), andbythe hypothefis, ^ : O = ^Ji : ^ ; /. ■ : ♦ CZ :D(B. s-pr-'is). /. ♦ Z3 D (B- 5- pr- io-)» or Q C ♦• Secondly, let ^ = |P , then will ^ ^ ^ . For ^ : O = : D (B. 5. pr. 7), and ^ : Q = : ^ (hyp.) ; .*. ■ : D= -V : ♦ (B. 5- pr- lO' and /. O = 4 (B. 5, pr. 9). Thirdly, if ^ 13 , then will O ZI ♦ ; becaufe C W ^"d : ^ = ^ : Q ; /. ^ C O, by the firft cafe, that is, Q 13 ^ . /. If the firft has the fame ratio, &c. BOOKV. PROP. XV. THEOR. 179 AGNITUDES /lave the fame ratio to one another which their equimultiples have. Let and be two magnitudes ; then, : ■ : : M' : M' ^ ^^ For A : = a .*. # : H :: 4 • : 4 • (B. 5- pr- 12)- And as the fame reafoning is generally applicable, we have # : ■ :: M' A : M'h. /, Magnitudes have the fame ratio, &c. i8o BOOKF. DEFINITION XIII. DEFINITION XIII. The technical term permutando, or alternando, by permu- tation or alternately, is ufed when there are four propor- tionals, and it is inferred that the firft has the fame ratio to the third which the fecond has to the fourth ; or that the firft is to the third as the fecond is to the fourth : as is Ihown in the following propofition : — Let# : 4 ::19 :B) by " permutando" or "alternando" it is inferred ^ : ^ •• ^ • B • It may be neceffary here to remark that the magnitudes A, ^j V7H7 muft be homogeneous, that is, of the fame nature or fimilitude of kind ; we muft therefore, in fuch cafes, compare lines with lines, furfaces with furfaces, folids with folids, &c. Hence the ftudent will readily perceive that a line and a furface, a furface and a folid, or other heterogenous magnitudes, can never ftand in the re- lation of antecedent and confequent. BOOK V. PROP. XVL THEOR. i8i F four magnitudes of the fame kind be proportionals, they are afo proportionals ivhen taken alternately. Let ^ : Q : : H : ▲ , then ip : B - U • ^ • ForM fl : M O :: ^ : Q (B. 5. pr. 15), d M ^ : M Q :: H : ^ (^yP-) ^nd (B. 5. pr. 11) alfo /;; m : /;; ▲ ' • H * ^ (^- 5- P''- ^ S) > .*. M ^ : M Q :: w : /« ^ (B. 5. pr. 14), and /. if M ^ C. =» or ^ zw B ? then will M Q C :=, or 33 ;« ^ (B. 5. pr. 14) ; therefore, by the fifth definition, .*. If four magnitudes of the fame kind, &c. 1 82 BOOK F. DEFIXmOX XFL DEFLS'ITIOX XVI. DnmxxDO, by di^ i :- . r - :h ere ire :": _ : r : : : ; - r , and it is inferred, l i : J-.e exceli : : : - : toood b to the fecood, £i iJie ev; ;::;::::; :r : -, b to tbe fenrth. le: : 3 ::C : D; far ** diridendo ** it b inferred A miners B : B : : C minns '^ : ~" . Ac; ; r :: . r :: -e,A b fbppt^i :: r-e rti ;' ^ : B, and C i:'- -" ; if thb be -:: ±: :i : :ut to have r : i :: ::..- :£ D greater iIjj: .2 : S :A :-. D :C; -A :A :: zuz^C :C. BOOK V. PROP. XVII. THEOR. 183 [F magnitudes, taken jointly, be proportionals, they Jhall alfo be proportionals ii-hen taken feparately : \, that is, if tivo magnitudes together have to one of them the fame ratio which two others have to one ofthefe, the remaining one of the fir ft two Jhall have to the other the fame ratio which the remaining one of the laft two has to the other of thefe. Let tp + CI: O ::" + ♦: ♦, then will ^ : O :: ■ : ♦. Take M ^ C « O to each add M Q, then we have M V + M Q C 'w O + M Q, orM(V + CI) C (^^ + M: D: but becaufe IP + 0:0::"+#: ♦ (hyp.), and M (IP + O) C (;« + xM) Q ; .-. M (■ + ♦) C (^^ + M) 4 (B. 5. def. 5) ; /. M ^ + M ♦[=//;♦+ M ♦ ; .'. M '^ C ^ ^ . by taking M ^ from both fides : that is, when 'SI ^ ^ m U, then M T~ m ^ . In the Tame manner it may be proved, that if M ^ r= or ^ OT U, then will M =r or — \ m ^ • and /. V : O : : ? : ♦ (B. 5. def. 5). .*. If magnitudes taken jointly, &c. l84 book V. DEFINITION XV. DEFINITION XV. The term componendo, by compofition, is ufed when there are four proportionals ; and it is inferred that the firft toge- ther with the fecond is to the fecond as the third together with the fourth is to the fourth. Let A : B : : : D ; then, by the term " componendo," it is inferred that A-|.B:B:: -j-D:D. By " invertion" B and O may become the firft and third, A and _ the fecond and fourth, as B : A : : D : C , then, by " componendo," we infer that B + A : A ; : D -|- . : ^ . BOOK F. PROP. XVIII. THEOR. i8s F magnitudes, taken feparately, be proportionals , they fliall alfo be proportionals when taken jointly : that is, if the Jirji be to the fecond as the third is to the fourth, the firji. and fecond together fhall be to the fecond as the third and fourth together is to the fourth. Let IP : O then fP + Q : Q for if not, let |p -f- Q fuppofing ^ • • ^^ • v^ • • but ^ : Q : : not = ^ ; • (B. 5. pr. 17); : ^ (hyp.); .'•■:#::■: 4 (B. 5. pr. n); •••• = ♦ (B. 5- pr- 9). which is contrary to the fuppofition ; .'. ^ is not unequal to ^ ; that is =: ^ ; *, If magnitudes, taken feparately, &c. B B i86 BOOK V. PROP. XIX. THEOR. F a isohole magnitude be to a whole, as a magnitude taken from the firji, is to a magnitude taken from the other ; the remainder ffoall be to the remainder, as the ivhole to the whole. Let l^ + O :■ + ♦:: IP :■, then will Q: ::'P> + 0:H+'', For tP + a : V :: ■ + t : ■ (^l^er.), .*. O : V ••: ♦ :■ (divid.), again Q : 4 ^^ 9 ^ H (alter.), butlP + 0:» + # ::^:B hyp.); therefore Q : : : ^ + D : ■ + ♦ (B. 5. pr. 11). ,*, If a whole magnitude be to a whole, &c. DEFINITION XVII. The term " convertendo," by converfion, is made ufe of by geometricians, when there are four proportionals, and it is inferred, that the firft is to its excefs above the fecond, as the third is to its excefs above the fourth. See the fol- lowing propofition : — BOOK V. PROP. E. THEOR. 187 F four magnitudes be proportionals, they are alfo proportionals by converjion : that is, the Jirjl is to its excefs above the fecond, as the third to its ex- cefs above the fourth. then fhall • O • • ^ • ■ '-W, Becaufe therefore ' .-. o •. #0: :0::B : (divid.), i :: ^ : ■ (inver.). (compo.). .'. If four magnitudes, &c. DEFINITION XVIII. " Ex squall " (fc. diflantia), or ex aequo, from equality of diftance : when there is any number of magnitudes more than two, and as many others, fuch that they are propor- tionals when taken two and two of each rank, and it is inferred that the firft is to the laft of the firft rank of mag- nitudes, as the firft is to the laft of the others : " of this there are the two following kinds, which arife from the different order in which the magnitudes are taken, two and two." i88 BOOK V. DEFINITION XIX. DEFINITION XIX. " Ex asquali," from equality. This term is ufed iimply by itfelf, when the firft magnitude is to the fecond of the firft rank, as the firft to the fecond of the other rank. ; and as the fecond is to the third of the firft rank, fo is the fecond to the third of the other ; and fo on in order : and the in- ference is as mentioned in the preceding definition; whence this is called ordinate proportion. It is demonftrated in Book 5. pr. 22. Thus, if there be two ranks of magnitudes, A, B, , . , E, F, the firft rank, and L, M, N , < ' , P, Q, the fecond, fuch that A : B : : L : M, B : :: M : , C : U : : .\ : ( ) , D : E : : o : P, E : F : : P : Q ; we infer by the term " ex squali" that A : F :: L :Q. BOOK F. DEFINITION XX. 189 DEFINITION XX. " Ex ^quali in proportione perturbata feu inordinata," from equality in perturbate, or diforderly proportion. This term is ufed when the firft magnitude is to the fecond of the firft rank as the laft but one is to the laft of the fecond rank ; and as the fecond is to the third of the firft rank, fo is the laft but two to the laft but one of the fecond rank ; and as the third is to the fourth of the firft rank, fo is the third from the laft to the laft but two of the fecond rank ; and fo on in a crofs order : and the inference is in the i8th definition. It is demonftrated in B. 5. pr. 23. Thus, if there be two ranks of magnitudes, A. , B , C , D , E , F , the firft rank, and , M , N , O , P , Q , the fecond, fuch that A : B : : P : Q , B : C : : O : P , C^ : D : : N : O , D : ' : : * : N , : : : : vr ; the term " ex xquali in proportione perturbata feu inordi- nata" infers that A : r : : ^ : <,> . 190 BOOK V. PROP. XX. THEOR. F i/iere be three magnitudes , and other three, which, taken two and two, have the fame ratio ; then, if the jirjl be greater than the third, the fourth fiall be greater than the fixth ; and if equal, equal ; and if lefs, lefs. Let ^, 0> J be the firft three magnitudes, and ^, Oj ^> be the other three, fuch that fp :0 ::4 :0,andC) :B ::0:#- Then, if ^ IZ> =» or Z] , then will ^ CI, =, orZl ^. From the hypothefis, by alternando, we have andO :0 ::■:•; .*. "P :♦::■: • (B. 5- pr- n); /. if I^F d, =, or Z] , then will ^ C =, orI3 (B. 5. pr. 14). ,*, If there be three magnitudes, 6cc. BOOK V. PROP. XXL THEOR. 191 F t/iere be three magnitudes, arid other three which have the fame ratio, taken two and two, but in a crofs order ; then if the fir ft magnitude be greater than the third, the fourth fliall be greater than the fixth ; and if equal, equal ; and if lefs, lefs. Let I, be the firft three magnitudes, and ^, O*, fpt, the other three, fuch that ^ : A : : O •# > ^"^ A ' H - • ^ ■ O ' Then, if f C. =. or ID ■, then will ♦ C =, Zl #. Firft, let ^ be C ■ : then, becaufe ^ is any other magnitude, ¥•*!=■' A (6. 5-pr.8); butO :#::¥: A (^yp-); .-. O :# !=■ :A (B. 5-pr- 13); and becaufe {^ : ■ :: ^ : (j (hyp.) ; and it was fliown that (^ '. % d H ' iil > .*. O : " C C : ♦ (B. 5- pr- 13); 192 BOOK F. PROP. XXI. THEOR. •• • =] ♦, that is ^ C I . Secondly, let ^ H ; then fhall ^ = ^. For becaufe ^ B, V :* = ■ :dl (B. 5.pr.7); but : il = 0> : (hyp.). and ^ * A = O : ^ (hyp- ^"'l ifiv.), .-. O : # = : ♦ (B. 5. pr. II), .-. ^ = i (B. 5. pr. 9). Next, let be Z3 ■? then ^ fhall be Z3 ; for B C and it has been (hown that (§ • ^ ^ ^ * ▼' and il : = : O; /. by the firft cafe is C ^j that is, ^ ^ 9 . /. If there be three, &c. BOOK V. PROP. XXII. THEOR. 193 F there be any number of magnitudes, and as nuuiy others, 'which, taken two and two in order, have the fame ratio ; the frji JJjall have to the lajl of the firft magnitudes the fame ratio which the frji of the others has to the lajl of the fame . N.B. — This is ifually cited by the words "ex (egua/i," or "ex cequo." irft, let there 36 magnitud es^ and as many others ▲ ,0 = ? fuch that w •• ♦ "♦ : 0, and ^ :il ::0 • > then fliall 1^ • ^ • • ▼ • ♦ = ■«. Let thefe magnitudes, as well as any equimultiples whatever of the antecedents and confequents of the ratios, lland as follows : — and M ^,« ♦, N ' , M ^, w <;>, N 1, becaufe |p : ^ : : ^ : O ? :: M ^ :/«<3 (B. 5. p. 4). For the fame reafon w ^ : N : ; /« <^ : N | ; and becaufe there are three magnitudes, c c /.Mm: m 194 BOOK F. PROP. XXII. THEOR. and other three, M ^ , w <^ , N , which, taken two and two, have the fame ratio ; /. ifMip CZ, =, ori:N B then will M ^ CZ. =. oi' Z] N , by (B. 5. pr. 20) ; and .*. ^ : ■ : : ^ : # (def. 5). Next, let there be four magnitudes, ^ , ^, H ^ ^ » and other four, ^ , ^, IB , ▲ , which, taken two and two, have the fame ratio, that is to fay, ^ • ^ • • O ' #' ♦ :■::•: , and A : ^ ::m : ▲, then fhall IP : ^ : : O * ^ ' for, becaufe l[p , ^^, , are tliree magnitudes, and <^ , ^f , other three, which, taken two and two, have the fame ratio ; therefore, by the foregoing cafe, ^^ : ■ : : (2> • ^, but a : 4 : • «■ : -^ ; therefore again, by the firfl; cafe, ip : ^ : : (^ '- ^ f and fo on, whatever the number of magnitudes be. ,*, If there be any number, &c. BOOK V. PROP. XXIII. THEOR. 195 F t/iere be afiy number of tnagnitudes, and as many others, ivhich, taken two and two in a crofs order, have the fame ratio ; the firjl fliall have to the laji of the firjl magnitudes the fame ratio which the firji of the others has to the laji of the fame. N.B. — This is ifually cited by the words " ^x aquali in proportione perturbatd ;" or " ex aquo perturbato." Firft, let there be three magnitudes, ^j(^> |) and other three, ' > O ' ^ » which, taken two and two in a crofs order, have the fame ratio ; o Let thefe magnitudes and their refpective equimuhiples be arranged as follows : — M ,M^,m^,M ,,,m(^,m%, then f IQ ::M ' : M Q (B. 5. pr. 15); and for the fame reafon but^ :q ::<2> :0 (hyp.). that is, |; : U ' :o and Q :■ : •♦ then fhall ^ :■ : = ♦ 196 BOOK V. PROP. XXIII. THEOR. .-. M ip :MQ ::<^ :# (B. 5. pr. n); and becaufe O : ■ : : ^ : <2> (Jiyp-)> .-. M Q : w H : : ^ : w ^ (B. 5. pr. 4) ; then, becaufe there are three magnitudes, M W, M Q, w ■, and other three, M , m (2), w ^ , which, taken two and two in a crofs order, have the fame ratio ; therefore, if M [^, ^, or "H ;;; J j then will M [Z, =r, or ;i] ;;/ (B. 5. pr. 21), and /. ,;; : ■ :: -J. : # (B. 5. def. 5). Next, let there be four magnitudes, and other four, (2)j ^j ■> A.? which, when taken two and two in a crofs order, have the fame ratio ; namely. IP :D :: ■ D ■ ::• andH • •# -0 en fhall "O For, becaufe ^^ ^, | are three magnitudes, BOOKF. PROP. XXIII. THEOR. 197 and 9, SI, i^, other three, which, taken two and two in a crofs order, have the fiime ratio, therefore, by the firfl cafe, ^ : H •• • ^^ but ■ : :: <^ : #, therefore again, by the firft cafe, y : ^ : : /S ' A ? and i'o on, whatever be the number of fuch magnitudes. .*. If there be any number, &c. 198 BOOK V. PROP. XXIV. THEOR. jF the firji has to the fecond the fame ratio which the third has to the fourth, and the fifth to the fecond the fame which the fix th has to the fourth, the fir fi and fifth together Jhall have to the fecond the fame ratio which the third and fix th together have to the fourth. First. Fifth. Second. D Third. Sixth. Fourth. Let ip : U: :a:<^, and (2> : D: :•:#. '+0 •Q: • ■ + • : 4 then For <2>:D--: #: ^ (%P-). and Q : ^ :: ^ : B (^yP-) ^"^ (invert.), .-. 0> •¥::#:■ (B- 5- Pr- 22); and, becaufe thefe magnitudes are proportionals, they are proportionals when taken jointly, .•• V+ 0:0:: •+ ■: • (B. 5- pr- 18), but o : D • : • • '- (hypO. .-. V + O : U ::#+■• t (B. 5- pr. 22). /. If the firft, &c. BOOK V. PROP. XXV. THEOR. 199 F four magnitudes of the fame kind are propor- tionals, the greatejl and leaf of them together are greater than the other two together. Let four magnitudes, ■ -j- ^, H -|- ■-' , |^, and |^ , of the fame kind, be proportionals, that is to fay, and let ■ -f- O ^^ ^^ greateft of the four, and confe- quently by pr. A and 14 of Book 5, ^ is the leaft ; then will ^+1314- beClB+ +D; becaufe If + Q :■+>:: O : ♦, but + Dl= ■ + (B. 5. pr. 19), (hyp.). .'. "f [= ■(B. 5. pr. A); to each of thefe add O "4" ^7 •*. fP + O + 1= ■ + o + ♦■ If four magnitudes, &c. 2o,o BOOK V. DEFINITION X. DEFINITION X. When three magnitudes are proportionals, the firfl is laid to have to the third the dupHcate ratio of that which it has to the fecond. For example, if A, b', C, be continued proportionals, that is, A : B :: B : C, A is faid to have to C the dupli- cate ratio of x\ : B ; or — r= the fquare of — . This property will be more readily feen of the quantities 'J ^"f , , J, tor /T !' '. u ' '.'. li ■ '• a \ and — ^ r^ r= the fquare of — = r. or of iJy f jr~ , for — ^ -3 = the fquare of — =:— . a r " ' DEFINITION XI. When four magnitudes are continual proportionals, the firft is faid to have to the fourth the triplicate ratio of that which it has to the fecond ; and fo on, quadruplicate, &c. increafing the denomination ftill by unity, in any number of proportionals. For example, let. A, B, C, D, be four continued propor- tionals, that is, A ; : : : : C :: C : D ; A is faid to have to D, the triplicate ratio of N to iJ ; or - := the cube of—. BOOK K DEFINITION XL 201 This definition will be better underftood, and applied to a greater number of magnitudes than four that are con- tinued proportionals, as follows : — Let^r", ' yar> ^y be four magnitudes in continued pro- portion, that is, ^ »■':': : '■ ar '-'-ar '• (i, . ar' „ , , -ar^ then =: r" r= the cube or — ^ r. a Or, let ar', ar*, ar^, ur', ar, a, be fix magnitudes in pro- portion, that is ar* : rtr* :: ar^ ■ ar* :: ar" : ar" :: ar' : ar :: ar : a, a r - a r then the ratio — = r" zrz the fifth power of — : zr: r. a ^ rtr* Or, let a, ar, ar^, ar^, ar*, be five magnitudes in continued proportion; then — 5 := -5 =z the fourth power of — ::=:-. DEFINITION A. To know a compound ratio : — When there are any number of magnitudes of the fame kind, the firfi: is faid to have to the lafl: of them the ratio compounded of the ratio which the firfl has to the fecond, and of the ratio which the fecond has to the third, and of the ratio which the third has to the fourth ; and fo on, unto the lafl; magnitude. For example, if A , B , C , D , be four magnitudes of the fame kind, the firft A is faid to have to the lafl: D the ratio compounded of the ratio of A to B , and of the ratio of B to C , and of the ratio of C to D ; or, the ratio of DD A B C D E F G H s K L 202 BOOKF. DEFINITION A. A to D is faid to be compounded of the ratios of \ to B , B to C , and c to |j. And if A has to B the fame ratio which 1 has to V , and B to C the fame ratio that G has to H, and C to D the fame that K has to L ; then by this definition, \ is said to have to L> the ratio compounded of ratios which are the fame with the ratios of E to F, G to H, and K to L. And the fame thing is to be underftood when it is more briefly exprefled by faying, \ has to D the ratio compounded of the ratios oft to F, G to H, and K to I . In like manner, the fame things being fuppofed ; if has to the fame ratio which \ has to D, then for fhort- nefs fake, is faid to have to the ratio compounded of the ratios of E to F, G to H, and K to L. This definition may be better underftood from an arith- metical or algebraical illuftration ; for, in fact, a ratio com- pounded of feveral other ratios, is nothing more than a ratio which has for its antecedent the continued produdl of all the antecedents of the ratios compounded, and for its confequent the continued produdl of all the confequents of the ratios compounded. Thus, the ratio compounded of the ratios of 2 : ;, 4 : 7, 6 : 1 1, 2 : 5, is the ratio of ; X X 6 X 2 : X X 1 1 X 5, or the ratio of 96 : 11 55, or -^2 : 385. And of the magnitudes A, B, C, D, E, F, of the fame kind, A : F is the ratio compounded of the ratios of A : B, B : C C : D, D : E, E : F ; for A X B X X X E : B X C X x E X F, ^^ nx'x XEXF = T' ""^ ^^^ ""^"^ °^ "^ '■ ^' BOOK r. PROP. F. THEOR. 203 ATIOS wAic/i are cojnpounded of the fame ratios are the fame to one another. Let A : B : : F : G, B : C :: G : H, C: D::H:K, and D : E :: K : L. A B C D E F G H K L Then the ratio which is compounded of the ratios of A : R, ^ : , : , : t , or the ratio of A : E, is the fame as the ratio compounded of the ratios of F : G, G : H, H : K, K : L, or the ratio of F : L. For ^ = F G' B C ~" G H' C __ D "■ H K' a„d^ = K AX XX F X X X X X -: X X X ■ — X L and /. - F — L or the ratio of A : E is the fame as the ratio of F : L. The fame may be demonflrated of any number of ratios fo circumftanced. Next, let A : B : : K : L, B: C:: H: K, C: D:: G: H, D: E :: F: G. 204 BOOK V. PROP. F. THEOR. Then the ratio which is compounded of the ratios of A : B, B : C, C : D, D : E, or the ratio of A : E, is the fame as the ratio compounded of the ratios of :L, : K, G : H, F : , or the ratio of F :L. For - = -, I and — =: — ; r. A X X X D . X X X F X ^^ X E — L X X X G * ^•■"^ •••! = -' F L or the ratio of A : ¥ is the fame as the ratio of F : L. ,", Ratios which are compounded, &c. BOOK V. PROP. G. THEOR. 205 F fever al ratios be the fame to fever al ratios, each to each, the ratio which is compounded of ratios which are the fame to the firft ratios, each to each, jhall be the fame to the ratio compounded of ratios which are the fame to the other ratios, each to each. A B C: D E ¥ G H P Q R S T a bed e f g h V w X y If A : B : : d : ^ and A : B : : P : Q a:b:: : \\ CD ::€ -.d C:D::Q: R c:d:: w : X E:F ::e:f E:F ::R S e:f:: X : Y and G : II :: g : A G:H:: S : T g:h:: Y : Z then P : T = ^ " • • p^^ P A a Z3 > 2 — ^' - i- R D d = > R __ E e S" — * F — 7 ^ 9 ^ G ff f H h ) and • '' X 9 X k X ■ __ ^""^ • • X R X s X r — X X X X X X ~ » and /. -p = - ~ > 01 rP : T = : '/.. If feveral ratios, &c. 2o6 BOOK V. PROP. H. THEOR. F a ratio which is compounded of fever al ratios be the fame to a ratio which is compounded of fever al other ratios ; and if one of the firjl ratios, or the ratio which is compounded of fever al of them, be the fame to one of the laji ratios, or to the ratio which is com- pounded of Jeveral of them ; then the remaining ratio ofthefirjl, or, if there be more than one, the ratio compounded of the re- maining ratios, JJi all be the fame to the remaining ratio of the la/i, or, if there be more than one, to the ratio compounded of thefe remaining ratios. A B C D E F G H F Q R S T X Let A : B, B : C, C : D, D : E, E : F, F : G, G : H, be the firft ratios, and P : Q^_Qj^R, R : S, S : T, T : X, the other ratios ; alfo, let A : H, which is compounded of the iirfl: ratios, be the fame as the ratio of P : X, which is the ratio compounded of the other ratios ; and, let the ratio of A : E, which is compounded of the ratios of A : B, B : C, C : D, D : E, be the fame as the ratio of P : R, which is compounded of the ratios P : Q,^ Qj R. Then the ratio which is compounded of the remaining firft ratios, that is, the ratio compounded of the ratios E : F, F : G, G : H, that is, the ratio of E : H, fhall be the fame as the ratio of R : X, which is compounded of the ratios of R : S, S : T, T : X, the remaining other ratios. Becaufe - BOOK V. PROP. H. THEOR. 207 , X f. X C X D X L X J: X & P X Q X R X S X I l; X L X D X E X F X G X H Q X R X s X 'I' X X' , X [; X C X 1^ w K X F X G P X Q w R X S x f f! X L X D X K •^ h- X(,XU — O X R ^ ^ X T X X' anH ^ X B X C X D _ P X Q ^ E XCX DX E — Qx R' , E X F X G R X - X I • * F X G X H i X I X X» /. E : H = R : X. ,*, If a ratio which, &c. 2o8 BOOK V. PROP. K. THEOR. F t/iere be any number of ratios, and any number of other ratios, fuch that the ratio which is com- pounded of ratios, which are the fame to the frji ratios, each to each, is the fame to the ratio which is compounded of ratios, which are the fame, each to each, to the lajl ratios — and if one of the firji ratios, or the ratio which is compounded of ratios, which are the fame to federal of the firjl ratios, each to each, be the fame to one of the lajl ratios, or to the ratio which is compounded of ratios, which are the fame, each to each, to fever al of the lajl ratios — then the re- maining ratio of the firjl ; or, if there be more than one, the ratio which is compounded of ratios, which are the fame, each to each, to the remaining ratios of the firJi, Jhall be the fame to the remaining ratio of the lajl ; or, if there be more than one, to the ratio which is compounded of ratios, which are the fame, each to each, to thefe remaining ratios. h k m n s AB, CD, EF, GH, K L, MN. a b c d e t g O P , O R , S T , V W , X Y , h k I m n p abed e i g Let A:B, C:D, E:F, G:H, K:L, M:N, be the firft ratios, and o :!', (^:R , ^ :T, V :W, X : , the other ratios ; and let A : B zn a '. b , C :D = b :c. E :F HZ L id. G :H ~~7. 'i : '' J K : L — e :f. M:N T"^ / I P" • BOOK F. PROP. K. THEOR. 209 Then, by the definition of a compound ratio, the ratio of ,7 In- is compounded of the ratios of j :/,, /; ic, c -Jt J ','> g :/"./"■?-. which are the fame as the ratio of A : B, C : D, E : F, G : H, K : L, M : N, each to each. Alfo, :^ ^; h •k. Q^ \R •=. k :/. : : T = /: m. V : VV = m : n. -- ; — n P- Then will the ratio oi h\p be the ratio compounded of the ratios of h:k, k:l, I '.my m'.n, n:p, which are the fame as the ratios of :p , Ct :R , S :T , V :W , X :Y , each to each. /, by the hypothefis a '•!? = h:p. Alfo, let the ratio which is compounded of the ratios of A: B, C : D, two of the firfl: ratios (or the ratios of j ict for \ : ^ = J : A, and C" : P = , : ), be the fame as the ratio of a : d, which is compounded of the ratios of a : b, b : c, c : d, which are the fame as the ratios of : , : , : , three of the other ratios. And let the ratios of h : s, which is compounded of the ratios of h : k, k : m, m : n, n : s, which are the fame as the remaining firft ratios, namely, E : F, G : H, K : L, M : N ; alfo, let the ratio of e : g, be that which is com- pounded of the ratios e : f, f : g, which are the fame, each to each, to the remaining other ratios, namely, V :W, X : Y . Then the ratio of h : s fhall be the fame as the ratio of e : g ; or h : s r= erg. p AXCXKX'.XKXM g X 6 X ,- X ,i X r X /• :-^ X 11 X F X H X I X X — - 6 X c X i X . X/ X y ' £ £ 2IO BOOK V. PROP. K. THEOR. , OX ox ^X . X ■ ^x^x^x?»x» ^ X R X I X V, X kXlXmXnXp by the compolition of the ratios ; . cX/XcXrfXfX; h X kX I XrnXn ''iXcXdXfXfXj kX I Xm X n Xp (hyp.). uXl w c X ^ X £ X / kX kX I w .7 X c ^ dX cX/Xg kX t Xm ^ n Xp' but — — A X C __ ^- X X . a Xb Xc __ h Xk X i . ;X. £XD ;X X bxcxd kX I Xm' , f X -J X f X ; _. »t X n ' ' i X e X fX i' n Xp' A„J C X ^" X t X • h X k X m X n ,, . And .v. V^vV — k VmVn Vs ("7?-). >?Xc X/X^ k X m X n X s and m X n n Xp — e T X f Xg (hyp.). • • • h X k X m k X m X n X X n . s — e f • • • s ^^ e /. h : s : — e : g- /, If there be any number, &c. ; * Al"-ebraical and Arithmetical expositions of the Fifth Book of Euclid are given m Hyrne's Doctrine of Proportion ; published by Williams and Co. London. 1841. BOOK VI. DEFINITIONS. I. ECTILINEAR figures are faid to be fimilar, when they have their fe- veral angles equal, each to each, and the fides about the equal angles proportional. II. Two fides of one figure are faid to be reciprocally propor- tional to two fides of another figure when one of the fides of the firft is to the fecond, as the remaining fide of the fecond is to the remaining fide of the firft. III. A STRAIGHT line is faid to be cut in extreme and mean ratio, when the whole is to the greater fegment, as the greater fegment is to the lefs. IV. The altitude of any figure is the straight line drawn from its vertex perpendicular to its bafe, or the bafe produced. 2;2 BOOK VI. PROP. I. THEOR. RIANGLES and parallelo- grams having the fame altitude are to one another as their bafes. Let the triangles 1 and m have a common vertex, and their bafes and in the fame ftraight hne. Produce i both ways, take fucceffively on — — produced lines equal to it ; and on — — — pro- duced lines succefTively equal to it ; and draw lines from the common vertex to their extremities. A The triangles j^-JKJt^ thus formed are all equal to one another, fmce their bafes are equal. (B. i . pr. 38.) A and its bafe are refpectively equi- i multiples of ■ and the bafe BOOK VL PROP. I. THEOR. 2- 3 ^ In like manner » _ and its bafe are refpec- i lively equimultiples of |^ and the bafe — — . .*. Ifm or 6 times ^ (^ := or 13 « or 5 times B then m or 6 times — — C ^ or ;^ « or 5 times u-m i , m and « ftand for every multiple taken as in the fifth definition of the Fifth Book. Although we have only fhown that this property exifts when m equal 6, and n equal 5, yet it is evident that the property holds good for every multiple value that may be given to m, and to n. a (B. 5. def 5.) Parallelograms having the fame altitude are the doubles of the triangles, on their bafes, and are proportional to them (Part i), and hence their doubles, the parallelograms, are as their bafes. (B. 5. pr. 15.) Q. E. D. 214 BOOK VI. PROP. II. THEOR. * F a Jlraight line be draivn parallel to any Jide ■——■■-> of a tri- angle, it fliall cut the other tides, or thoj'e Jides produced, into pro- portional fegments . And if any Jlraight line ^— ^— divide the fides of a triangle, or thofe fides produced, into proportional feg- ments, it is parallel to the remaining fide —■•■-■■■■■», Let PART I. I, then {hall • •• ' ■•• Draw and and (B. I. pr. 37); ■V- V \- ^ : ! \ (B.5.pr.7);but (B. 6. pr. 1), (■■■■■■■■■* • I • «■■■■«■■■« * IttlllllBlB*, (B. 5. pr. II). BOOK VI. PROP. 11. THEOR. 21 PART II. Let Let the fame conftrudlion remain. becaulc : :: 1/ : • \ > (B.6. pr. I); and ■ »«aiaflVB»« ■ « • • • but «■■■ ^aai^ i\ / -Z= (hyp.), ), : i \ (B. 5. pr. 1 1 .) :. (B. 5- pr- 9) ; ■■■■■■•. , and at the but they are on the fame bafe -■■■■■•■i fame fide of it, and •°. II (B- i.pr. 39) Q. E. D. 2l6 BOOK VI. PROP. III. THEOR. RIGHT /ine ( ) bifeSling the angle of a triangle, divides the op- pofite Jide into fegments — — ™) proportional to the conterminous Jides (- )• And if a Jiraight line (- — ) drawn from any angle of a triangle divide the oppojite Jide ( ■■■■■■) into fegments ( , ....■■■■..) proportional to the conterminous fides (— — — , ■ ), it bifeSls the angle. PART I. Draw -■ to meet then, ^ ^ (B. i.pr. 29), ^ = < ; but ^ = 1 , .-. ^ =-# , .*. .......... = I I (B. I. pr. 6); and becaufe ■ ■■■■■■■IHB (B. 6. pr. 2) ; but (B. 5. pr. 7). BOOK FI. PROP. III. THEOR. 217 PART II. Let the fame conftrudtion remain, and (B. 6. pr. 2) ; but — ^.^ : ......... :: «-..—. ; ...^^ (hyp.) ■■.■■■• (B. 5. pr. .1). and ,*, ■.■■•■■■a. zzz ■ (B. c. pr. o), and /. ^ = ^(B. I. pr. 5); but fince II «..; ^ _ ^^ and ^ = ^ (B. i. pr. 29); /. ^ =y, and =: ^, and .*. ........ bifedls ^ Q^E. D. F F «^i8 BOOK VI. PROP. IV. THEOR. N equiatigular tri- angles ( ^ \ and ,-•'* \ ) the fides about the equal angles are pro- portional, and the Jides which are ^L oppojite to the equal angles are homologous. Let the equiangular triangles be fo placed that two fides ^^ and oppofite to equal angles ^^^ may be conterminous, and in the fame ftraight line; and that the triangles lying at the fame fide of that flraight line, may have the equal angles not conterminous, ofite to jtKk , and fl^ to j^|^ . 1. e. opp Draw -■•■■••■••t and ', Then, becaufe ▲ = ▲ II and for a like reafon, •■•■—■•■• - - „ / ;B.i.pr.28); ■ ""•"5 is a parallelogram. But (B. 6. pr. 2) ; BOOK FI. PROP. IF. THEOR. 219 and lince ' ^ — — — (B. i. pr. 34), ^_i. : .......«i.. * and by • • alternation, — ^— ^ : — — — ^ j: -—.-««— ; — (B. 5. pr. 16). In like manner it may be fhown, that ■^i^HMiMaBH^ ■«»•«■«■«»« ^^B^i^ia^^^ iiMiJiiaiiia* and by alternation, that •■■^^■^■Ma o^^^^^^^ JJ •■■■■■■•■^ J ■■■■■■■■■■* but it has been already proved that ^a^^^^^^^m * ^MMHHWHHMM * ■■■■■■■■■■« and therefore, ex squali. ■■■■■■■■■« * ■■■■■■■■■■■ (B. 5. pr. 22), therefore the fides about the equal angles are proportional, and thofe which are oppolite to the equal angles are homologous. Q. E. D. 220 BOOK VI. PROP. V. THEOR. F tivo triangles have their Jides propor- tional (•■■•■•■■- : ■■«.■■■■» :: ^mmmmmmm, \ ) and (' ■■■■■■■■■■■» * «•■■■■«■« :: — — ^— : i^^»— .) they are equiangular, and the equal angles are fubtended by the homolo- gous fides. From the extremities of , draw and , making W= iB (B. i.pr. 2.1;); = (B. I. pr. 32) , and fince the triangles are equiangular. (B. 6. pr. 4); but (hyp.); and confequently (B. 5. pr. 9). In the like manner it may be fhown that BOOK VI. PROP. V. THEOR. 221 Therefore, the two triangles having a common bafe — «^— , and their fides equal, have alfo equal angles op- A =^a„d^ = ^ polite to equal fides, i. e. (B. I. pr. 8). But ^F = j^^ (conft.) and .*• jj^^ =: M/^ ^ for the fame reafon ^^^ z= flU? ^"^ confequently ^^ := (B. i. 32); and therefore the triangles are equiangular, and it is evi- dent that the homologous fides fubtend the equal angles. CUE. D. 222 BOOK VI. PROP. VI. THEOR. 4 .A F tivo triangles ( _^**^ _^*'^ and .^^___^ ) have one angle ( wKk ) o/' ///<' one, equal to one , angle ( m \ ) of the other, and the Jides ^ about the equal angles proportional, the HPI^ triangles Jhall be equiangular, and have thofe angles equal which the homologous Jides fubt end. From the extremities of of Z:^ about ^— , one of the fides ■ \ , draw and maki in g ▼ =A and ^W zz. ^=4 then ^ =: (B. I. pr. 32), and two triangles being equiangular. ■ ■^■■■•■■■a (B. 6. pr. 4) ; but •••■>••••••• (hyp.); (B. 5. pr. 11), and confequently •«■■«■«•■•* (B. 5. pr. 9); k BOOK VI. PROP. VI. THEOR. 223 .*. -^ \ = >,♦* in every refpedl. (B. I. pr. 4). But ^yf = j^ (conft.), and /. ZLj ■=. J^ ; and fince alio ■ \ ::z: JHl , ^\ = -^ (B. i.pr. 32); j/\ and .*. A*I,....dW ^"d -^ \ are equiangular, with their equal angles oppolite to homologous Tides. Q^E. D. 224. BOOK VI. PROP. VII. THEOR. A /\ F two triangles ( A and A » * * ) Aave one angle in • each equal ( ' equal to ^^ ), the \ Jides about two other angles proportional \ (-^ : — - :: : ..-—), ^l A and each of the remaining angles ( and ^..^ ) either lefs or not lefs than a right angle, the triangles are equiangular, and thofe angles are equal about which the Jides are proportional. Firft let it be alTumed that the angles ^^ and <.^ are each lefs than a right angle : then if it be fuppofed that i^A ^"'^ ^^ contained by the proportional fides, are not equal, let ^\ be the greater, and make Becaufe ^ = ^ (hyp.), and ^\ = ^J (conft.) /. ^», = ^^--B ^B. I. pr. 32); \ BOOK VI. PROP. FIT. THEOR. 225 (B. 6. pr. 4), but — ^^— : ■ :: — — ■■— : — .— .. (hyp.) • 9 (B. 5. pr. 9), 4 and ,*. ^^ = ^^ (B. 1. pr. 5). But ^^B is lefs than a right angle (hyp.) ,•, ^^^ is lefs than a right angle ; and ,', ^B muft be greater than a right angle (B. i. pr. 13), but it has been proved ^ '^^^.^ and therefore lefs than a right angle, which is abfurd. ,*. ^<^ and ^-\ are not unequal ; .', they are equal, and fince ^B rz / \ (hyp.) 4=4 (B. I. pr. 32), and therefore the tri- angles are equiangular. -^ and ^5 But if '^^ and ^*-^ be aflumed to be each not lefs than a right angle, it may be proved as before, that the triangles are equiangular, and have the fides about the equal angles proportional. (B. 6. pr. 4). Q. E. D. OG 226 BOOK VI. PROP. Fin. THEOR. N a right angled triangle .j^S^ 9 ^^ ( triangle and to each other. Becaufe ^^p» ~ common to i a perpendicular ( be drawn from the right angle to the oppojitejide, the triangles ) on each Jide of it are fimilar to the whole (B. I. ax. 1 1 ), and and t\^< ;B. I. pr. 32); and ..^^^^l are equiangular ; and conlequently have their Tides about the equal angles pro- portional (B. 6. pr. 4), and are therefore limilar (B. 6. def. I). In like manner it may be proved that ^^ is fimilar to k ; but has been ihewn to be limilar to and k are fimilar to the whole and to each other. Q. E. D. BOOK VI. PROP. IX. PROB. 22: ROM a given Jiraig/it line { " '" ) to cut off any required part . From either extremity of the given line draw — ^■—"••••■t. making any angle with ■ ; and produce ■•••••> till the whole produced line ■mtiBH* contains ■ > as often as -■■"■"■- contains the required part. Draw , and draw ■ is the required part of For fi nee * ■«■■•••« (B. 6. pr. 2), and by compolition (B. 5. pr. 18) ; — ^— > -■".- mmmmmm but mmm '•■--•■ contains as often as contains the required part (conft.) ; ■■■— is the required part. Q. E. D. 228 BOOK VI. PROP. X. PROB. and draw O divide a Jlraight line ( ) fanilarly to a given divided line )• From either extremity of the given line — i^ draw ■■■■««a>s»aaaKaj»«M making any angle ; take and >■•••«« equal to refpedlively (B. i. pr. 2) ; and draw — — --— and -— II to it. or and Since ( — j are II, (B.6. pr. 2), (B. 6. pr. 2), and ,*, the given line fimilarlv to (conft.), (conft.). is divided Q.E. D. BOOK VL PROP. XL PROB. 229 O yf«i/ a third proportional to two given Jlraight lines At either extremity of the given line ^— i^— » draw .---—— making an angle ; take ....... .^ = , and draw I : make .,.._... =: , and draw || — ^ • (B. I. pr. 31.) l u - is the third proportional to -■^— ^^ and _ For fince but (B. 6pr. 2); ■(conft.) ; (B. 5. pr. 7). Q^E. D. 230 BOOK VI. PROP. XII. PROB. O find a fourth pro- portional to three given lines Draw and take and alfo draw and making any angle ; (B. r. pr. 31); is the fourth proportional. bU. (;■. On account of the parallels, (B. 6. pr. 2); •} = { = .} (conft.); ■ ■■•■■■■•« • (B. 5. pr. 7). Q^E. D. BOOK VI. PROP. XIII. PROP. ^31 O Jind a mean propor- tional between two given Jlraight lines { «ia«MMl«a«ni«l } Draw any ftraight line make — and bifed and from the point of bifedtion as a centre, and half the line as a radius, defcribe a femicircle draw — ^— ^— JL — — — cs is the mean proportional required. Draw and Since '^^^ is a right angle (B. 3. pr. 31), and ^^^^— is J_ from it upon the oppofite fide, •*. •^^"■^ is a mean proportional between — and ' (B. 6. pr. 8), and .*. between — — — and — •• (conft.). Q. E. D 232 BOOK VI. PROP. XIV. THEOR. QJJ A L parallelograms \ and •which have one angle in each equal, have the Jides about the equal angles reciprocally proportional ( ■)• II. And parallelograms which have one angle in each equal, and the fides about them reciprocally proportional, are equal. Let and - and ; and and ^■^~", be fo placed that ' ■■■ ' ■ -■— may be continued right Unes. It is evi- dent that they mayaflume this pofition. (B. i. prs. 13, 14, 1 5-) Complete % Since •V \ \ \ (B. 5. pr. 7.) BOOK VI. PROP. Xir. THEOR. 233 (B. 6. pr. I.) The fame conftrudtion remaining : r A (B. 6. pr. I.) — (hyp.) (B. 6. pr. I.) (B. 5. pr. II.) and .*. ^Hi^ = ^^ (B. 5. pr. 9). Q^E. D. H H 234 BOOK VI. PROP. XV. THEOR. I. QUAL triangles, which have one angle in each equal ( ^^ ^ ^B ), have the JiJt's about the equal angles reciprocally proportional (■ -- )• II. j^i And two triangles which have an angle of the one equal to an angle of the other, and the Jides about the equal angles reci- procally proportional, are equal. Let the triangles be {o placed that the equal angles ^^ and ^A may be vertically oppolite, that is to lay, lb that ^mmmmi^m and — -^— may be in the lame ftraight line. Whence alfo i and -aiM^MMM mull be in the fame ftraight line. (B. i. pr. 14.) Draw ■— — — , then > (B. 6. pr. I.) (B. 5. pr. 7.) (B. 6. pr. I.) BOOK VI. PROP. XV. THEOR. 235 > (B. 5. pr. II.) II. Let the fame conftruction remain, and (B. 6. pr. I.) and A (B. 6. pr. I.) But — — . : ^— :: ; . ■ , (hyp.) (B.5 pr. 11); (B. 5. pr. 9.) • • • > -^ Q.E. D. 236 BOOK VI. PROP. XVI. THEOR. PART I. Y four Jh'ciight lines be proportional the reSlangle ( ■ : ■■). !■>..._.. ) contaified by the extremes, is equal to the rectangle X .........) contained by the means. PART II. And if the reSt- angle contained by the extremes be equal to the reBangle con- tained by the means, the four Jlraight lines are proportional. PART I. From the extremities of •— i» and " ^M^BHB and ————— _L to them and ^ draw and ——•.-—- refpedlively : complete the parallelograms ^^^^H and I And fince, • BB««»»«B» (hyp.) (conft.) • H (B. 6. pr. 14), BOOK VI. PROP. XVI. THEOR. 237 that is, the redtangle contained by the extremes, equal to the redangle contained by the means. PART II. Let the fame conftrudlion remain ; becaufe • • (' and 11 ^. -....-..■. , (B. 6. pr. .4). But = , and — — i^ ^ — — — . ^conft.) (B. 5. pr. 7). Q. E. D. 238 BOOK VI. PROP. XVII. THEOR. fince then PART I F three Jlraight lines be pro- portional ( — 1 : ^^^mmt :: — — : ) the reSlangle under the extremes is equal to the fquare of the mean. PART II. And if the reSlangle under the ex- tremes be equal to the fquare of the mean, the three fir aight lines are proportional. PART I. Aflume X and X (B. 6. pr. 16). or But X ■"9 - X _.«i» ^ ; therefore, if the three ftraight hnes are proportional, the redlangle contained by the extremes is equal to the fquare of the mean. Aflume • PART II. « , then X • « • • and m» 9 ^^ • (B. 6. pr. 16), • • wmm Q. E. D. BOOK VI. PROP. XVIIL THEOR. 239 N a given Jlraight line ( ) ^,. to conftruSi a reBilinear figure fimilar to a given one ( and /imiiarly placed. ^ Relblve the given figure into triangles by drawing the lines -«—--- and ••..••*•. At the extremities of — — — ^ make ^ = Jb^^ and % = \J^ : again at the extremities of ■ and ^^ = ^^\ : in like manner make ? = ^ ^"^ V = V • make -^^ =: Then -v is fimilar to It is evident from the conftrudlion and (B. 1. pr. 32) tliat the figures are equiangular ; and fince the triangles W ^" w are equiangular; then by (B. 6.pr.4), and 240 BOOK VI. PROP. XVIII. THEOR. Again, becaule ^^^ and ^^B are equiangular. mm ** tt»afBffia«ai • /. ex asquali. (B. 6. pr. 22.) In like manner it may be fhown that the remaining fides of the two figures are proportional. .-. by (B. 6. def. i .) is fimilar to and fimilarly fituated ; and on the given line Q^E. D. BOOK VI. PROP. XIX. THEOR. 241 I M I L A R trian- gles ( A and ^^^^^k ) are to one another in the duplicate ratio of their homologous Jides. Let ^^ and A be equal angles, and and homologous fides of the fimilar triangles i^HHft and MKKKL and ^^^^^^ ^nd on -.-.-.—— the greater of thefe lines take --■— ■ a third proportional, fo that * «■■■■■■■■■ \ draw (B. 6. pr. 4) ; but (B. 5. pr. 16, alt.), ■■■>■■■ MiSa (conll:.), — confe- 1 1 242 BOOK VI. PROP. XIX. THEOR. A\ quently ^^^ rz ^^ for they have the fides about the equal angles ^^ and ^Ik reciprocally proportional (B. 6. pr. 15); Aa-A\ (B. 5 pr. 7); ^^^L : ^^ :: .... (B. 6. pr. I), that is to fay, the triangles are to one anotlier in tlie dupli- cate ratio of their homologous fides — i— and i^— -i (B. 5. def. 11). Q^ E. D. BOOK FI. PROP. XX. THEOR. 243 IMILAR poly- gons may be di- vided into the fame number of fimilar triangles, eachfimilar pair of ivhic/i are propor- tional to the polygons ; and the polygons are to each other in the duplicate ratio of their homologous fides . Draw and and and " 5 refolving the polygons into triangles. Then becaufe the polygons are limilar, and — ■■■•■■««««■ and ♦=♦ are fimilar, and ^^ ^ ^J (B. 6. pr.6); but ^F^ = w becaufe they are angles of fimilar poly gons ; therefore the remainders ^^ and ^k hence nmmmmmmmmm * ■■>..■«•• \\ -_..._-__ * on account of the fimilar triangles, are equal ; * ? 244 BOOK VI. PROP. XX. THEOR. and --. : :: I on account of the fimilar polygons, ■ ■■■■*■■■■» • _^.^_^_— — ___ •• ■■■■■■■HMM* • _^ ex asquali (B. 5. pr. 22), and as thefe proportional fides contain equal angles, the triangles s ^^^ and ^^^ are fimilar (B. 6. pr. 6). In like manner it may be fhown that the triangles ^^F and ^^K are fimilar. ^^ and ^^m But -^^» is to ^^^m in the duplicate ratio of ..-■■..... to .—.—>— (B. 6. pr. 19), and ^^^ is to ^^ in like manner, in the duplicate ratio of -.■•■•■■-.• to —-——.; >> (B. 5-P'-. II); Again ^^^^ is to ^^^ in the duplicate ratio of ^^^ to ^^ to — ^— — , and ^^^F is to ^^r in T BOOK VL PROP. XX. THEOR. 245 the duplicate ratio of ^i^— to . and as one of the antecedents is to one of the confequents, fo is the fum of all the antecedents to the fum of all the confequents ; that is to fay, the fimilar triangles have to one another the fame ratio as the polygons (B. 5. pr. 12). But ^^M is to ^^^F in the duplicate ratio of to Q ED 246 BOOK VL PROP. XXI. THEOR. ECTILINEAR Jigures ( <?«</ which are fimi/ar to the fame Jigure ( are fimilar alfo to each other. Since HHiBll^ and are fimi- lar, they are equiangular, and have the fides about the equal angles proportional (B. 6. def. i); and fince the figures and '^%. are alfo fimilar, they are equiangular, and have the fides about the equal angles proportional ; therefore IHIBl^ and l^Hhk. are alfo equiangular, and have the fides about the equal angles pro- portional (B. 5. pr. 1 1), and are therefore fimilar. Q,E. D. BOOK VI. PROP. XXII. THEOR. 247 PART I. Y four Jlraight lines be pro- portional (^^^ I ^^— :: — : ), the Jiinilar reSiilinear figures fimilarly described on them are aljo pro- portional. PART II. And if four fimilar reSlilinear figures, fimilarly defcribed on four jlraight lines, be proportional, the firaight lines are alfo proportional. Take and — to fince PART I. a third proportional to , and —■••••■•• a third proportional — > and — — — > (B.6.pr. ii); :: ; (hyp.), ■ — ■ :: -— — : -■••••••••• (conft.) .*. ex asquali. but and (B. 6. pr. 20), 248 BOOK VI. PROP. XXII. THEOR. and ,*. (B. 5. pr. 11). PART II. Let the fame conftrudlion remain (B. 5. pr. II). (hyp-). (conft.) (^E. D. BOOK VI. PROP. XXIII. THEOR. 249 QUIANGULAR parallel- ograms ( and ^m^ ) are to one another in a ratio compounded of the ratios of their fdes. Let two of the fides and -«... . about the equal angles be placed fo that they may form one ftraight line. Since ▼ 4. M -- f\\ , and 1^^ = ^W (hyp.). and .*. + and form one flraight line (B. I. pr. 14) ; complete ^ , Since # • ■■«■• and # (B. 6. pr. i), (B.6. pr. i). has to - to .„. a ratio compounded of the ratios of , and of ^^— — to — n^— »» . K K Q^E. D. 250 BOOK FT. PROP. XXIV. THEOR. -B N any parallelogram ( ^7 / ) the parallelograms ( r^i and ^ I ) 'which are about the diagonal are Jimilar to the whole, and to each other. B-J ^^ As ^ I and ^ I have a common angle they are equiangular ; but becaufe and are fimilar (B. 6. pr. 4), and the remaining oppofite fides are equal to thofe, , It 1 and fn I have the fides about the equal angles proportional, and are therefore fimilar. In the fame manner it can be demonftrated that the parallelograms ^7 / and ^ / are fimilar. Since, therefore, each of the parallelograms B ..^E ^. is fimilar to ^1 I , they are fimilar to each other. Q. E. D. BOOK VI. PROP. XXF. PROB. 251 O defcribe a reSlilinear Jigure, ivhic/i /Jiall be Jimilar to a given reBilinear Jigure ( equal to another (^^ ). ), and Upon defcribe i_ defcribe | | = ^^, and upon «_ and having ^M ^ (B. I. pr. 45), and then smm Between and ■H»» will lie in the fame flraight line (B. I. prs. 29, 14), and nu»»H.. find a mean proportional (B. 6. pr. 13), and upon _^««_«i defcribe Jtt^ 9 iimilar to and fimilarly fituated Then For fince and are fimilar, and (confl.), ■ ■■■■■■■■■ (B. 6. pr. 20) ; 252 BOOK FI. PROP. XXV. PROP. but 1 • • but .^d^k = ■ (B.6.piM); (B.5.pr.ii); and .♦. (conft.), (B. 5. pr. 14); I and (conft.) ; confequently. which is limilar to is alfo = Q. E. D. BOOK VI. PROP. XXVI. THEOR. 253 F fitnilai' and Jimilarly pojited parallelograms have a common angle, they are about the fame diagonal. For, if poffible, let be the diagonal of draw ■ (B. I. pr. 31). Since P.. ^ are about the fame and have diagonal ^^^^^^^^ , ana nave jmm common, they are fimilar (B. 6. pr. 24) ; but - (hyp.). and .*. (B. 5. pr. 9.), which is abfurd. 3 is not the diagonal of in the fame manner it can be demonftrated that no other line is except : . Q. E. D. 254 BOOK VI. PROP. XXVII. THEOR. F al/ the reBangles contained by the fegments of a given Jlraight line, the greateji is the fquare which is defer ibed on ha f the line. be the unequal fegments, equal fegments ; For it has been demonftrated already (B. 2. pr. 5), that the fquare of half the line is equal to the redlangle con- tained by any unequal fegments together with the fquare of the part intermediate between the middle point and the point of unequal fection. The fquare defcribed on half the line exceeds therefore the redtangle contained by any un- equal fegments of the line. Q.E. D. BOOK VI. PROP. XXFIII. PROP. ^SS O divide a given Jlraight line fo that the rec- tcuigle cojitained by its segments may be equal to a given area, not exceeding the fquare of half the line. Let the given area be := Bifedl — or make and if But if muft ■ ■■■ » **** ymmmm*** ■■■1 •; . 2 the problem is folved. 9 ■II ^ -4;- ••■»■■■■••■ 9 then ■■■ (hyp.). Draw make - with ^-i ■ ■■■■IMHM I or as radius defcribe a circle cutting the given line ; draw Then •••— ^ wiMMaBB.-a- ^ (B. 2. pr. 5.) = .2 But + (B. I. pr. 47); 256 BOOK VI. PROP. XXVIII. PROB. .\ X — ■ + ^— = ' + \ from both, take — i"— ^— ^^ and ""■■ X ■— — ^^—•■••« S3 «MMB But " ■■ ' ■ ' =: — — — •• (conft.), and /. ■-■"" is fo divided that •"•"• X ——————— ^: —•••.-2^ Q^E. D. BOOK VI. PROP. XXIX. PROB. ^S7 O produce agivenjlraight line ( ), fo that the reBangle con- tained by the fegments between the extremities of the given line and the point to which it is pro- duced, may be equal to a given area, i. e. equal to the fquare on Make draw -—"--■ draw with the radius meeting Then —■■-—' •-, and But and 5 and ', defcribe a circle ■ produced. ' (B. 2. pr. 6.) = — -^ + (B.i.pr.47.) := the given area. Q^E. D. L L 258 BOOK VI. PROP. XXX. PROB. 1 O cut a given finite Jlraight line ( ■ — ■• ) in extreme and mean ratio. On defcribe the fquare I (B. I. pr. 46) ; and produce X " , fo that ^ s (B. 6. pr. 29); take and draw ^ meeting ^ Then U ■•aatB •■ X- ;B. I. pr. 31; ■ ■■■taaBB and is .*. ^ n ; and if from both thefe equals be taken the common part \ I , which is the fquare of ■ will be = ■ , which is = '■■- X ■ ■««■■«■*« * that is and .■ is divided in extreme and mean ratio. (B. 6. def. 3). CLE. D. BOOK FI. PROP. XXXI. THEOR. 2 59 F any fimilar reSlilinear figures be fimilar ly defer ibed on the fides of a right an- gled triangle ( ^''•»^ ), the figure defer ibed on the fide (■....i ) fuh- tending the right angle is equal to the futn of the figures on the other fides. From the right angle draw to •> m. then ■■■■■•■■■MHB : ^_— i— perpendicular but (B. 6. pr. 8). (B. 6. pr, 20). (B. 6. pr. 20). • •ammmmmm%mmm Hence but + + and /. Q. E. D. 26o BOOK VI. PROP. XXXII. THEOR F two triangles ( ^ ^ ^«^ /^\ ), have two fides pro- portional ( ..1.^.^^ : I I \\ .. .......... I •.•••..•••.), and be fo placed \ i7^ an angle that the homologous Jides are pa- rallel, the remaining Jides ( one right line. and ) form Since = (B. I. pr. 29); and alfo fince -^— ^ || ••■••••■>■ = ^^ (B. I- pr. 29); = ^^^ ; and fince ■ ■■■«■■■««■ • — (hyp.). the triangles are equiangular (B. 6. pr. 6) ; M = /S but A+ +A = + A + JI = ■*■«■»■»« I I 1 (B. I. pr. 32), and /. -*••—■» and lie in the fame ftraight line (B. i. pr. 14). Q,E. D BOOKVL PROP. XXXIII. THEOR. 261 N egua/ circles ( O-O ), angles. whether at the centre or circumference, are in the fame ratio to one another as the arcs on which they Jland ( fo alfo are fedlors. i-J::- o )■> Take in the circumference off 1 any number of arcs "■— ■ , ■— , &c. each ^ ^m» ^ and alfo in the circumference of f j take any number of arcs • , , Sec. each ^ •***«*•, draw the radii to the extremities of the equal arcs. Then fince the arcs — , —. , i..., &c. are all equal, the angles # , # , ^, &c. are alfoequal (B. 3. pr.27); .*. ^V is the fame multiple of which the arc is of ^1^ • and in the fame nianner ^Bi^ is the fame multiple o is of the arc which the arc .... •>•• V* 262 BOOK VI. PROP. XXXIII. THEOR. Then it is evident (B. 3. pr. 27), if" ^11^ (or if m times w ) C> => ^ Mfg^ I (or n times ^ ) then ^^fc_i,i«^^ (or »; times '••••^) C!> ■ .....^••* (or n times ) ; .... , (B. 5. def. 5), or the angles at the centre are as the arcs on which they fland ; but the angles at the circumference being halves of the angles at the centre (B. 3. pr. 20) are in the fame ratio (B. 5. pr. 15), and therefore are as the arcs on which they ftand. It is evident, that fedlors in equal circles, and on equal arcs are equal (B. i. pr. 4; B. 3. prs. 24, 27, and def. 9). Hence, if the fedors be fubftituted for the angles in the above demonftration, the fecond part of the propofition will be eftablifhed, that is, in equal circles the fedlors have the fame ratio to one another as the arcs on which they ftand. Q^E. D. BOOK VI. PROP. A. THEOR. Y the right line {'mmm^um,), bifeSling an external angle ^H of the tri- yf 263 angle z. meet the oppojite ^ Jide (-^^— •) produced, that whole produced fide ( "■■•), and its external fegment (——--—) will be proportional to the fides (-^— — ■..— and ), which contain the angle adjacent to the external bifeSled angle. For if I be drawn || -.---»•»■• ^ then ^^ = \ / , (B. i. pr. 29) ; and = ^,(hyp-). = ^P, (B. I. pr. 29); r........ zz. ■ I III. , (B. I. pr. 6), and (B. 5. pr. 7) ; But alfo. ■ •■■■■■■■■■■ , (B. 6. pr. 2); and therefore (B. 5. pr. I,). Q. E. D. 264 BOOK VI. PROP. B. THEOR. X F an angle of a triangle be bi- Je5ied by a Jlraight line, which likewife cuts the bafe ; the rec- tangle contained by the Jides of the triangle is equal to the rectangle con- tained by the Jegments of the bafe, together with the fquare of the Jlraight line which bifedls the angle. Let be drawn, making ^ = ^; then fhall X + ^r \ defcribe I About y \ defcribe J (B. 4. pr. 5), produce 1 ■■ to meet the circle, and draw ■■■»>— Since ^ = ^^ (hyp-)' and ^^ = ^ (B. 3. pr. 21), • * •C*i»l»l ind \ are equiangular (B. i. pr. 32) ; (B. 6. pr. 4) ROOK FI. PROP. B. THEOR. 265 (B. 6. pr. 16.) .- X + ' (B. 2. pr. 3); ■*«■«■ but X — — = X (B- 3- pr. 35)' X = X Q.E. D. MM 266 BOOK VI. PROP. C. THEOR. fhall Y from any angle of a triangle a Jlraight line be drawn perpendi- cular to the bafe ; the rectangle contained by the fdes of the tri- angle is equal to the reSlangle contained by the perpendicular and the diameter of the circle defcribed about the triangle. From draw ■«>ii»«afa«« .. X of ^y ..-• — ; then Xthe diameter of the defcribed circle. Defcribe O (B. 4. pr. 5), draw its diameter and . and draw ^-im— • then becaufe ^ ■ >• (conft. and B. 3. pr. 31) ; ,Xl = /> (B. 3. pr. 21); .%*< is equiangular to / ^ • MHHaHMMHM l/^ and ,*. .-— .— (B. 6. pr. 16). (B. 6. pr. 4); X Q^E. D. BOOK VI. PROP. D. THEOR. 267 IHE reStangle contained by the ' diagonals of a quadrilateral figure I infcribed in a circle, is equal to \ both the reBangles contained by its oppoftte Jides. / /be any quadrilateral / o fieure infcribed in and draw and then X X Make ^k = W (B.i.pr. 23), ^ = ^ ; and (B. 3. pr. 21); = «■■■■■■!■■« (B. 6. pr. 4); and ,*. X X (B. 6. pr. 16) ; again, becaufe ^^ ^ ^F (conft.), X- ■■■■■•■•• 268 BOOK FL PROP. D. THEOR. and\/ = \^ (B. 3. pr. 21); •■«■■■«•■■• * ■■■■■*>■■■ THE END. (B. 6. pr. 4); and ,'. '•"••■-••• ^ .^__^ ^ .•■■•■•■■■■• ^ a^^^MB (B. 6. pr. 16) ; but, from above, X = X ; — = X + X (B. 2. pr. I . Q^E. D. cHiswirK: PRiNirn by c. " iirxTiNoinM. fuciJ ■O.,' ^ \'yj